L(s) = 1 | − 2.70·2-s − 1.27·3-s + 5.31·4-s + 5-s + 3.43·6-s − 0.968·7-s − 8.95·8-s − 1.38·9-s − 2.70·10-s + 11-s − 6.75·12-s + 1.17·13-s + 2.61·14-s − 1.27·15-s + 13.5·16-s − 1.19·17-s + 3.73·18-s + 4.63·19-s + 5.31·20-s + 1.23·21-s − 2.70·22-s + 3.13·23-s + 11.3·24-s + 25-s − 3.18·26-s + 5.57·27-s − 5.14·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s − 0.734·3-s + 2.65·4-s + 0.447·5-s + 1.40·6-s − 0.366·7-s − 3.16·8-s − 0.460·9-s − 0.855·10-s + 0.301·11-s − 1.95·12-s + 0.326·13-s + 0.700·14-s − 0.328·15-s + 3.39·16-s − 0.289·17-s + 0.880·18-s + 1.06·19-s + 1.18·20-s + 0.268·21-s − 0.576·22-s + 0.652·23-s + 2.32·24-s + 0.200·25-s − 0.624·26-s + 1.07·27-s − 0.972·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5295461964\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5295461964\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 1.27T + 3T^{2} \) |
| 7 | \( 1 + 0.968T + 7T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 1.19T + 17T^{2} \) |
| 19 | \( 1 - 4.63T + 19T^{2} \) |
| 23 | \( 1 - 3.13T + 23T^{2} \) |
| 29 | \( 1 - 9.52T + 29T^{2} \) |
| 31 | \( 1 + 7.86T + 31T^{2} \) |
| 37 | \( 1 + 4.17T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 5.79T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 - 9.54T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 6.06T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 2.72T + 97T^{2} \) |
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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
−8.659141571045240602999530554555, −7.87386037511174753065602059722, −6.92632209289088224339117311757, −6.57862792318716793209465568163, −5.80813247561650458596208461687, −5.07061285303226608012345734735, −3.40052831292719600927793527324, −2.62497746782770193079821118549, −1.49690050913096112494870746616, −0.60367772616461096826747229149,
0.60367772616461096826747229149, 1.49690050913096112494870746616, 2.62497746782770193079821118549, 3.40052831292719600927793527324, 5.07061285303226608012345734735, 5.80813247561650458596208461687, 6.57862792318716793209465568163, 6.92632209289088224339117311757, 7.87386037511174753065602059722, 8.659141571045240602999530554555