| L(s) = 1 | + 10·3-s − 10·5-s + 54·7-s + 57·9-s + 52·11-s − 104·13-s − 100·15-s + 76·17-s + 176·19-s + 540·21-s + 102·23-s + 25·25-s + 170·27-s − 612·29-s + 244·31-s + 520·33-s − 540·35-s + 176·37-s − 1.04e3·39-s − 260·41-s − 228·43-s − 570·45-s + 716·47-s + 1.52e3·49-s + 760·51-s − 808·53-s − 520·55-s + ⋯ |
| L(s) = 1 | + 1.92·3-s − 0.894·5-s + 2.91·7-s + 19/9·9-s + 1.42·11-s − 2.21·13-s − 1.72·15-s + 1.08·17-s + 2.12·19-s + 5.61·21-s + 0.924·23-s + 1/5·25-s + 1.21·27-s − 3.91·29-s + 1.41·31-s + 2.74·33-s − 2.60·35-s + 0.782·37-s − 4.27·39-s − 0.990·41-s − 0.808·43-s − 1.88·45-s + 2.22·47-s + 4.44·49-s + 2.08·51-s − 2.09·53-s − 1.27·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.76180921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.76180921\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 54 T + 199 p T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 10 T + 43 T^{2} - 10 p T^{3} - 20 p^{2} T^{4} - 10 p^{4} T^{5} + 43 p^{6} T^{6} - 10 p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 52 T - 546 T^{2} - 30576 T^{3} + 5191915 T^{4} - 30576 p^{3} T^{5} - 546 p^{6} T^{6} - 52 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 p T + 758 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 76 T - 1182 T^{2} + 217968 T^{3} - 4101293 T^{4} + 217968 p^{3} T^{5} - 1182 p^{6} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 176 T + 9602 T^{2} - 1347456 T^{3} + 197337611 T^{4} - 1347456 p^{3} T^{5} + 9602 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 102 T - 16333 T^{2} - 245106 T^{3} + 440245812 T^{4} - 245106 p^{3} T^{5} - 16333 p^{6} T^{6} - 102 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 306 T + 69019 T^{2} + 306 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 244 T - 11762 T^{2} - 2858704 T^{3} + 2422264147 T^{4} - 2858704 p^{3} T^{5} - 11762 p^{6} T^{6} - 244 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176 T - 27386 T^{2} + 7558144 T^{3} - 728630693 T^{4} + 7558144 p^{3} T^{5} - 27386 p^{6} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 130 T + 57499 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 114 T - 36287 T^{2} + 114 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 716 T + 185646 T^{2} - 85464624 T^{3} + 42930840883 T^{4} - 85464624 p^{3} T^{5} + 185646 p^{6} T^{6} - 716 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 808 T + 202542 T^{2} + 123274944 T^{3} + 83429108827 T^{4} + 123274944 p^{3} T^{5} + 202542 p^{6} T^{6} + 808 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1016 T + 395202 T^{2} + 229916736 T^{3} + 151653560251 T^{4} + 229916736 p^{3} T^{5} + 395202 p^{6} T^{6} + 1016 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 222 T - 222607 T^{2} + 40419762 T^{3} + 11217809916 T^{4} + 40419762 p^{3} T^{5} - 222607 p^{6} T^{6} - 222 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 p T - 526261 T^{2} + 114618 p T^{3} + 200313260492 T^{4} + 114618 p^{4} T^{5} - 526261 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 296 T + 268774 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 724 T + 110098 T^{2} - 263504144 T^{3} - 183787011197 T^{4} - 263504144 p^{3} T^{5} + 110098 p^{6} T^{6} + 724 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 1128 T + 116138 T^{2} - 191949504 T^{3} + 466482422019 T^{4} - 191949504 p^{3} T^{5} + 116138 p^{6} T^{6} - 1128 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 338 T + 402817 T^{2} + 338 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 326 T - 265431 T^{2} + 338463306 T^{3} - 433736482556 T^{4} + 338463306 p^{3} T^{5} - 265431 p^{6} T^{6} - 326 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 20 p T + 2737734 T^{2} + 20 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52791584734460086622153162514, −7.45857326576455870761936862825, −7.09576081949187992550250724651, −6.91948287385983987940943171891, −6.55555092320743951822947931025, −5.92638639544682722690857425334, −5.84382533891497879151812309225, −5.53839646680158800077460114233, −5.13886157948777042135485607024, −5.01600386529198311255931754368, −4.90585323296158568604891547174, −4.48658319887766172726082077117, −4.20358033488769796228077094421, −4.03114299727611983321161100009, −3.86893856190758692245190563057, −3.24921219672427592847655550351, −3.15720226148662213806430770761, −2.99283115741274949086605671450, −2.55642223348385093729272141952, −1.98493481368194412659782137440, −1.90180308293537425160887516981, −1.51032542988435912543212640059, −1.37852193725916278199585821531, −0.906773934654166703136206718463, −0.34494266497557071929616067106,
0.34494266497557071929616067106, 0.906773934654166703136206718463, 1.37852193725916278199585821531, 1.51032542988435912543212640059, 1.90180308293537425160887516981, 1.98493481368194412659782137440, 2.55642223348385093729272141952, 2.99283115741274949086605671450, 3.15720226148662213806430770761, 3.24921219672427592847655550351, 3.86893856190758692245190563057, 4.03114299727611983321161100009, 4.20358033488769796228077094421, 4.48658319887766172726082077117, 4.90585323296158568604891547174, 5.01600386529198311255931754368, 5.13886157948777042135485607024, 5.53839646680158800077460114233, 5.84382533891497879151812309225, 5.92638639544682722690857425334, 6.55555092320743951822947931025, 6.91948287385983987940943171891, 7.09576081949187992550250724651, 7.45857326576455870761936862825, 7.52791584734460086622153162514
