| L(s) = 1 | + 2·3-s + 32·4-s + 243·9-s + 1.42e3·11-s + 64·12-s − 5.76e3·19-s + 6.25e3·25-s + 1.45e3·27-s + 2.84e3·33-s + 7.77e3·36-s + 4.17e4·41-s + 2.25e4·43-s + 4.55e4·44-s − 3.36e4·49-s − 1.15e4·57-s + 1.45e5·59-s − 3.27e4·64-s − 6.71e4·67-s − 1.00e5·73-s + 1.25e4·75-s − 1.84e5·76-s + 2.90e3·81-s − 8.54e4·97-s + 3.45e5·99-s + 2.00e5·100-s + 4.64e4·108-s + 9.62e5·121-s + ⋯ |
| L(s) = 1 | + 0.128·3-s + 4-s + 9-s + 3.54·11-s + 0.128·12-s − 3.66·19-s + 2·25-s + 0.382·27-s + 0.454·33-s + 36-s + 3.88·41-s + 1.85·43-s + 3.54·44-s − 2·49-s − 0.469·57-s + 5.44·59-s − 64-s − 1.82·67-s − 2.21·73-s + 0.256·75-s − 3.66·76-s + 0.0491·81-s − 0.922·97-s + 3.54·99-s + 2·100-s + 0.382·108-s + 5.97·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.517604665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.517604665\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2^2$ | \( 1 - p^{5} T^{2} + p^{10} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T - 239 T^{2} - 2 p^{5} T^{3} + p^{10} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 - 474 T + p^{5} T^{2} )^{2}( 1 - 474 T + 63625 T^{2} - 474 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 1914 T + 2243539 T^{2} - 1914 p^{5} T^{3} + p^{10} T^{4} )( 1 + 1914 T + 2243539 T^{2} + 1914 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2882 T + 5829825 T^{2} + 2882 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{5} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 13926 T + p^{5} T^{2} )^{2}( 1 - 13926 T + 78077275 T^{2} - 13926 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 - 22550 T + p^{5} T^{2} )^{2}( 1 + 22550 T + 361494057 T^{2} + 22550 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 48486 T + p^{5} T^{2} )^{2}( 1 - 48486 T + 1635967897 T^{2} - 48486 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 67186 T + p^{5} T^{2} )^{2}( 1 - 67186 T + 3163833489 T^{2} - 67186 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{5} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 50402 T + 467290011 T^{2} + 50402 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{5} T^{2} + p^{10} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 89298 T + 4035092161 T^{2} - 89298 p^{5} T^{3} + p^{10} T^{4} )( 1 + 89298 T + 4035092161 T^{2} + 89298 p^{5} T^{3} + p^{10} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 7218 T + p^{5} T^{2} )^{2}( 1 + 7218 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 85450 T + p^{5} T^{2} )^{2}( 1 - 85450 T - 1285637757 T^{2} - 85450 p^{5} T^{3} + p^{10} T^{4} ) \) |
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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
−9.993252489793456584299168695219, −9.364064421178765124826338723949, −9.128498370841829531110678998681, −8.857838555633353646569381406057, −8.850292364926323428582517597495, −8.425006551889607456415549684671, −7.997690727559923816334350026253, −7.33120119805322704151136221899, −7.29448229970059844308554052645, −6.80713527577230047221331732441, −6.57154242186691608347509741169, −6.51308451208156845906228025216, −6.04243042793749869712498886843, −5.93158057794869630945499069994, −5.09555420621402732554174559827, −4.41117537509778747182293966677, −4.23492546532234285554245044412, −4.08408213424440842267020516906, −3.86016343296943946427710899201, −2.91366191153633931510454446880, −2.45564919940376800252853649021, −2.13392564217581523689071972740, −1.39917972284551575093876861804, −1.21716875341406124253350772264, −0.58935850413005895260933129088,
0.58935850413005895260933129088, 1.21716875341406124253350772264, 1.39917972284551575093876861804, 2.13392564217581523689071972740, 2.45564919940376800252853649021, 2.91366191153633931510454446880, 3.86016343296943946427710899201, 4.08408213424440842267020516906, 4.23492546532234285554245044412, 4.41117537509778747182293966677, 5.09555420621402732554174559827, 5.93158057794869630945499069994, 6.04243042793749869712498886843, 6.51308451208156845906228025216, 6.57154242186691608347509741169, 6.80713527577230047221331732441, 7.29448229970059844308554052645, 7.33120119805322704151136221899, 7.997690727559923816334350026253, 8.425006551889607456415549684671, 8.850292364926323428582517597495, 8.857838555633353646569381406057, 9.128498370841829531110678998681, 9.364064421178765124826338723949, 9.993252489793456584299168695219
