| L(s) = 1 | − 4·4-s + 42·9-s − 28·11-s + 12·16-s − 72·19-s + 52·29-s − 240·31-s − 168·36-s − 656·41-s + 112·44-s − 98·49-s + 2.46e3·59-s + 672·61-s − 288·64-s − 3.80e3·71-s + 288·76-s − 2.02e3·79-s − 7·81-s + 432·89-s − 1.17e3·99-s − 1.35e3·101-s − 964·109-s − 208·116-s − 738·121-s + 960·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 14/9·9-s − 0.767·11-s + 3/16·16-s − 0.869·19-s + 0.332·29-s − 1.39·31-s − 7/9·36-s − 2.49·41-s + 0.383·44-s − 2/7·49-s + 5.43·59-s + 1.41·61-s − 0.562·64-s − 6.36·71-s + 0.434·76-s − 2.88·79-s − 0.00960·81-s + 0.514·89-s − 1.19·99-s − 1.33·101-s − 0.847·109-s − 0.166·116-s − 0.554·121-s + 0.695·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1394373979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1394373979\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T^{2} + p^{2} T^{4} + p^{8} T^{6} + p^{12} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 14 p T^{2} + 1771 T^{4} - 14 p^{7} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 7474 T^{2} + 23538787 T^{4} - 7474 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 10658 T^{2} + 231811 p^{2} T^{4} - 10658 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 36 T + 10170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 404 T^{2} - 254476346 T^{4} - 404 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 26 T + 47795 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 120 T - 1618 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 29164 T^{2} + 3053796342 T^{4} - 29164 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 8 p T + 133986 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 265732 T^{2} + 29670935254 T^{4} - 265732 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 284346 T^{2} + 37502712587 T^{4} - 284346 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 528404 T^{2} + 114028658614 T^{4} - 528404 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 616 T + p^{3} T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 336 T + 458858 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1175116 T^{2} + 525951899734 T^{4} - 1175116 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 854236 T^{2} + 376944814630 T^{4} - 854236 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 1014 T + 1120119 T^{2} + 1014 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 776460 T^{2} + 702812559638 T^{4} - 776460 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 302190 T^{2} + 960976713683 T^{4} + 302190 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−8.758874945010871124384702912915, −8.624255205988623097960181033632, −8.566241642016171058844163407854, −7.933834704143804790678720291842, −7.77298554987503961263365679156, −7.40419676289149172650288038104, −7.02987950355812412862414971156, −7.00198318539079095395752752894, −6.77644020452773905440791942022, −6.40188160329373191011090938787, −5.68141460145288017547113342330, −5.67687892204942882487254397911, −5.52710301275790288475786828140, −4.85389613113856134963068168195, −4.82352917790929012541746536107, −4.23483860846907235139540127449, −4.13944779435838243460918755144, −3.77159704298825971939449878680, −3.40688579798447119643912695672, −2.67352045577803876477612793986, −2.66115853286060713733204180563, −1.74516262062128442405345671298, −1.64671238278383611027339063637, −1.03237242944832544739333277365, −0.083734173178597046867991202572,
0.083734173178597046867991202572, 1.03237242944832544739333277365, 1.64671238278383611027339063637, 1.74516262062128442405345671298, 2.66115853286060713733204180563, 2.67352045577803876477612793986, 3.40688579798447119643912695672, 3.77159704298825971939449878680, 4.13944779435838243460918755144, 4.23483860846907235139540127449, 4.82352917790929012541746536107, 4.85389613113856134963068168195, 5.52710301275790288475786828140, 5.67687892204942882487254397911, 5.68141460145288017547113342330, 6.40188160329373191011090938787, 6.77644020452773905440791942022, 7.00198318539079095395752752894, 7.02987950355812412862414971156, 7.40419676289149172650288038104, 7.77298554987503961263365679156, 7.933834704143804790678720291842, 8.566241642016171058844163407854, 8.624255205988623097960181033632, 8.758874945010871124384702912915
