| L(s) = 1 | − 2.18·2-s + 3-s + 2.78·4-s + 1.88·5-s − 2.18·6-s − 4.23·7-s − 1.72·8-s + 9-s − 4.11·10-s − 2.82·11-s + 2.78·12-s − 3.44·13-s + 9.26·14-s + 1.88·15-s − 1.80·16-s − 17-s − 2.18·18-s + 4.03·19-s + 5.24·20-s − 4.23·21-s + 6.18·22-s + 7.09·23-s − 1.72·24-s − 1.45·25-s + 7.54·26-s + 27-s − 11.8·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 0.577·3-s + 1.39·4-s + 0.841·5-s − 0.893·6-s − 1.60·7-s − 0.608·8-s + 0.333·9-s − 1.30·10-s − 0.852·11-s + 0.804·12-s − 0.956·13-s + 2.47·14-s + 0.486·15-s − 0.451·16-s − 0.242·17-s − 0.515·18-s + 0.924·19-s + 1.17·20-s − 0.924·21-s + 1.31·22-s + 1.47·23-s − 0.351·24-s − 0.291·25-s + 1.48·26-s + 0.192·27-s − 2.23·28-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 | \( 1 + 2.18T + 2T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 3.44T + 13T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 7.09T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 6.06T + 31T^{2} \) |
| 37 | \( 1 - 0.948T + 37T^{2} \) |
| 41 | \( 1 + 1.88T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 - 7.81T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 - 2.15T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 1.47T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 6.59T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 7.74T + 97T^{2} \) |
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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
−7.47028618541136584090999728777, −7.11666257466965311423577795731, −6.44177643409391029073421888630, −5.58637930412769233671714875762, −4.78400157902750369135126516374, −3.49264380799499570627556095831, −2.69719048384592717949130006722, −2.27038228582018395980958525070, −1.06274911894937543024746736046, 0,
1.06274911894937543024746736046, 2.27038228582018395980958525070, 2.69719048384592717949130006722, 3.49264380799499570627556095831, 4.78400157902750369135126516374, 5.58637930412769233671714875762, 6.44177643409391029073421888630, 7.11666257466965311423577795731, 7.47028618541136584090999728777
