| L(s) = 1 | − 2-s − 8·3-s + 8·4-s − 5·5-s + 8·6-s − 23·8-s + 27·9-s + 5·10-s − 12·11-s − 64·12-s + 156·13-s + 40·15-s + 23·16-s − 94·17-s − 27·18-s + 40·19-s − 40·20-s + 12·22-s − 32·23-s + 184·24-s − 156·26-s − 136·27-s − 100·29-s − 40·30-s − 248·31-s − 184·32-s + 96·33-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.53·3-s + 4-s − 0.447·5-s + 0.544·6-s − 1.01·8-s + 9-s + 0.158·10-s − 0.328·11-s − 1.53·12-s + 3.32·13-s + 0.688·15-s + 0.359·16-s − 1.34·17-s − 0.353·18-s + 0.482·19-s − 0.447·20-s + 0.116·22-s − 0.290·23-s + 1.56·24-s − 1.17·26-s − 0.969·27-s − 0.640·29-s − 0.243·30-s − 1.43·31-s − 1.01·32-s + 0.506·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7062731502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7062731502\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T - 7 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 8 T + 37 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 12 T - 1187 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 94 T + 3923 T^{2} + 94 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 40 T - 5259 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 32 T - 11143 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 p T + 33 p^{2} T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 434 T + 137703 T^{2} - 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 402 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 68 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 536 T + 183473 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T - 148393 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 560 T + 108221 T^{2} + 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 278 T - 149697 T^{2} + 278 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 164 T - 273867 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 672 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 82 T - 382293 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 1000 T + 506961 T^{2} - 1000 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 448 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 870 T + 51931 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1026 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48461318494159389155874356254, −11.32214359720826854236933398746, −11.07792168151889708641736234411, −10.85325408237148069869802194420, −10.28879787088962017465920663954, −9.315124222862868783132641762697, −9.114322345179221157374114619060, −8.373744728639186873744518239618, −8.049122645122926740322429390443, −7.24965668349914367154604446188, −6.54209981962817035793146821131, −6.48204303165828524738065310309, −5.71888782554540399015889674154, −5.66241769212158098904653499083, −4.65002191011998160768414083592, −3.65131788695460108162673006502, −3.52484088219002190307273225663, −2.18074891281495520238771493021, −1.40047529753773470494530497659, −0.40680845133924256923488009545,
0.40680845133924256923488009545, 1.40047529753773470494530497659, 2.18074891281495520238771493021, 3.52484088219002190307273225663, 3.65131788695460108162673006502, 4.65002191011998160768414083592, 5.66241769212158098904653499083, 5.71888782554540399015889674154, 6.48204303165828524738065310309, 6.54209981962817035793146821131, 7.24965668349914367154604446188, 8.049122645122926740322429390443, 8.373744728639186873744518239618, 9.114322345179221157374114619060, 9.315124222862868783132641762697, 10.28879787088962017465920663954, 10.85325408237148069869802194420, 11.07792168151889708641736234411, 11.32214359720826854236933398746, 11.48461318494159389155874356254
