| L(s) = 1 | − 1.85·2-s + 3-s + 1.44·4-s − 0.427·5-s − 1.85·6-s + 2.91·7-s + 1.03·8-s + 9-s + 0.793·10-s + 3.71·11-s + 1.44·12-s − 5.72·13-s − 5.41·14-s − 0.427·15-s − 4.80·16-s + 17-s − 1.85·18-s + 3.37·19-s − 0.617·20-s + 2.91·21-s − 6.89·22-s + 1.24·23-s + 1.03·24-s − 4.81·25-s + 10.6·26-s + 27-s + 4.21·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.577·3-s + 0.721·4-s − 0.191·5-s − 0.757·6-s + 1.10·7-s + 0.365·8-s + 0.333·9-s + 0.250·10-s + 1.12·11-s + 0.416·12-s − 1.58·13-s − 1.44·14-s − 0.110·15-s − 1.20·16-s + 0.242·17-s − 0.437·18-s + 0.773·19-s − 0.137·20-s + 0.636·21-s − 1.47·22-s + 0.259·23-s + 0.210·24-s − 0.963·25-s + 2.08·26-s + 0.192·27-s + 0.796·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 5 | \( 1 + 0.427T + 5T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 19 | \( 1 - 3.37T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 - 8.39T + 31T^{2} \) |
| 37 | \( 1 + 6.95T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 + 2.01T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 0.434T + 59T^{2} \) |
| 61 | \( 1 - 3.44T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 4.55T + 73T^{2} \) |
| 79 | \( 1 - 12.8T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 + 4.13T + 89T^{2} \) |
| 97 | \( 1 + 9.39T + 97T^{2} \) |
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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
−7.75043522891304740004938313396, −7.13288524829006739303624144071, −6.53492963776077682829897453935, −5.16555519167552574830822795948, −4.74240840466922507541370276815, −3.86984770537910289749699897257, −2.86862678012151028070992865343, −1.79536500389070032772552312179, −1.37961485730768765455483445672, 0,
1.37961485730768765455483445672, 1.79536500389070032772552312179, 2.86862678012151028070992865343, 3.86984770537910289749699897257, 4.74240840466922507541370276815, 5.16555519167552574830822795948, 6.53492963776077682829897453935, 7.13288524829006739303624144071, 7.75043522891304740004938313396
