L(s) = 1 | + 22.7·2-s + 81·3-s + 5.71·4-s − 1.39e3·5-s + 1.84e3·6-s − 3.35e3·7-s − 1.15e4·8-s + 6.56e3·9-s − 3.18e4·10-s + 3.56e4·11-s + 462.·12-s − 1.47e5·13-s − 7.63e4·14-s − 1.13e5·15-s − 2.65e5·16-s + 6.51e4·17-s + 1.49e5·18-s + 6.99e5·19-s − 7.98e3·20-s − 2.71e5·21-s + 8.10e5·22-s + 9.96e5·23-s − 9.33e5·24-s + 1.17e3·25-s − 3.36e6·26-s + 5.31e5·27-s − 1.91e4·28-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 0.577·3-s + 0.0111·4-s − 1.00·5-s + 0.580·6-s − 0.527·7-s − 0.994·8-s + 0.333·9-s − 1.00·10-s + 0.733·11-s + 0.00643·12-s − 1.43·13-s − 0.530·14-s − 0.577·15-s − 1.01·16-s + 0.189·17-s + 0.335·18-s + 1.23·19-s − 0.0111·20-s − 0.304·21-s + 0.737·22-s + 0.742·23-s − 0.574·24-s + 0.000603·25-s − 1.44·26-s + 0.192·27-s − 0.00588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(2.560001714\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.560001714\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 81T \) |
| 59 | \( 1 + 1.21e7T \) |
good | 2 | \( 1 - 22.7T + 512T^{2} \) |
| 5 | \( 1 + 1.39e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.56e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.47e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.51e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.99e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.96e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.88e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.43e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.85e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.87e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.51e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.39e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.41e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.22e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 7.34e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.06e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.11e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.97e8T + 7.60e17T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48508863188760930774824147190, −9.771529444101089863856444524498, −9.121035659088070452976706435644, −7.78414030147428607502583371139, −6.91704730441407557577765068186, −5.47856008268203536363432191437, −4.35428518172429676911507559893, −3.55832479712604342348656766642, −2.62942764462855449818874889803, −0.64504251349180628997191570251,
0.64504251349180628997191570251, 2.62942764462855449818874889803, 3.55832479712604342348656766642, 4.35428518172429676911507559893, 5.47856008268203536363432191437, 6.91704730441407557577765068186, 7.78414030147428607502583371139, 9.121035659088070452976706435644, 9.771529444101089863856444524498, 11.48508863188760930774824147190