| L(s) = 1 | + 2.46·2-s + 4.06·4-s + 1.39·5-s − 2.95·7-s + 5.07·8-s + 3.44·10-s − 5.47·11-s + 0.289·13-s − 7.27·14-s + 4.37·16-s − 0.545·17-s + 19-s + 5.68·20-s − 13.4·22-s + 2.62·23-s − 3.04·25-s + 0.714·26-s − 11.9·28-s − 10.4·29-s + 2.38·31-s + 0.620·32-s − 1.34·34-s − 4.13·35-s − 2.38·37-s + 2.46·38-s + 7.10·40-s + 4.74·41-s + ⋯ |
| L(s) = 1 | + 1.74·2-s + 2.03·4-s + 0.625·5-s − 1.11·7-s + 1.79·8-s + 1.08·10-s − 1.65·11-s + 0.0804·13-s − 1.94·14-s + 1.09·16-s − 0.132·17-s + 0.229·19-s + 1.27·20-s − 2.87·22-s + 0.547·23-s − 0.608·25-s + 0.140·26-s − 2.26·28-s − 1.94·29-s + 0.428·31-s + 0.109·32-s − 0.230·34-s − 0.698·35-s − 0.391·37-s + 0.399·38-s + 1.12·40-s + 0.740·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 5 | \( 1 - 1.39T + 5T^{2} \) |
| 7 | \( 1 + 2.95T + 7T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 0.289T + 13T^{2} \) |
| 17 | \( 1 + 0.545T + 17T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 + 2.38T + 37T^{2} \) |
| 41 | \( 1 - 4.74T + 41T^{2} \) |
| 43 | \( 1 + 4.18T + 43T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.78T + 59T^{2} \) |
| 61 | \( 1 - 5.06T + 61T^{2} \) |
| 67 | \( 1 - 1.98T + 67T^{2} \) |
| 71 | \( 1 - 9.03T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 17.1T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.32202673893143408348860149891, −6.37072623029735620520061163455, −6.02217960295647798273772893729, −5.30119821838589045999445827626, −4.86883656025740655205808279668, −3.82026692272367079425187614896, −3.20429601153307451226004398411, −2.58390610593005815947943138955, −1.82476939187803197517379599737, 0,
1.82476939187803197517379599737, 2.58390610593005815947943138955, 3.20429601153307451226004398411, 3.82026692272367079425187614896, 4.86883656025740655205808279668, 5.30119821838589045999445827626, 6.02217960295647798273772893729, 6.37072623029735620520061163455, 7.32202673893143408348860149891
