| L(s) = 1 | + 0.584·2-s + 0.172·3-s − 1.65·4-s + 5-s + 0.101·6-s − 2.74·7-s − 2.13·8-s − 2.97·9-s + 0.584·10-s − 0.429·11-s − 0.286·12-s − 6.91·13-s − 1.60·14-s + 0.172·15-s + 2.06·16-s + 5.29·17-s − 1.73·18-s − 1.63·19-s − 1.65·20-s − 0.474·21-s − 0.251·22-s − 2.26·23-s − 0.369·24-s + 25-s − 4.04·26-s − 1.03·27-s + 4.55·28-s + ⋯ |
| L(s) = 1 | + 0.413·2-s + 0.0996·3-s − 0.828·4-s + 0.447·5-s + 0.0412·6-s − 1.03·7-s − 0.756·8-s − 0.990·9-s + 0.184·10-s − 0.129·11-s − 0.0826·12-s − 1.91·13-s − 0.429·14-s + 0.0445·15-s + 0.515·16-s + 1.28·17-s − 0.409·18-s − 0.375·19-s − 0.370·20-s − 0.103·21-s − 0.0535·22-s − 0.473·23-s − 0.0754·24-s + 0.200·25-s − 0.793·26-s − 0.198·27-s + 0.860·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 0.584T + 2T^{2} \) |
| 3 | \( 1 - 0.172T + 3T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 + 0.429T + 11T^{2} \) |
| 13 | \( 1 + 6.91T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 1.63T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 - 3.93T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 7.74T + 37T^{2} \) |
| 41 | \( 1 + 5.18T + 41T^{2} \) |
| 43 | \( 1 + 6.95T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 + 6.35T + 59T^{2} \) |
| 61 | \( 1 - 9.00T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 5.08T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 + 6.64T + 83T^{2} \) |
| 97 | \( 1 - 0.828T + 97T^{2} \) |
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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
−10.29777119814374604027065380840, −9.808889374787792923180294679578, −8.996792902482164941132280721425, −8.013126328078941374500555446304, −6.77553088318695637853055378469, −5.65340940362821267541754816365, −5.02484126305243650496672456040, −3.56544450618704653806895186022, −2.63148264583210902853824114277, 0,
2.63148264583210902853824114277, 3.56544450618704653806895186022, 5.02484126305243650496672456040, 5.65340940362821267541754816365, 6.77553088318695637853055378469, 8.013126328078941374500555446304, 8.996792902482164941132280721425, 9.808889374787792923180294679578, 10.29777119814374604027065380840
