| L(s) = 1 | − 2.82·2-s + 2.95·3-s + 5.95·4-s − 8.33·6-s + 2.82·7-s − 11.1·8-s + 5.73·9-s − 0.493·11-s + 17.5·12-s + 1.57·13-s − 7.97·14-s + 19.5·16-s + 3.39·17-s − 16.1·18-s − 1.17·19-s + 8.35·21-s + 1.39·22-s − 0.0257·23-s − 32.9·24-s − 4.44·26-s + 8.09·27-s + 16.8·28-s − 4.46·29-s − 6.43·31-s − 32.7·32-s − 1.45·33-s − 9.58·34-s + ⋯ |
| L(s) = 1 | − 1.99·2-s + 1.70·3-s + 2.97·4-s − 3.40·6-s + 1.06·7-s − 3.94·8-s + 1.91·9-s − 0.148·11-s + 5.08·12-s + 0.437·13-s − 2.13·14-s + 4.88·16-s + 0.824·17-s − 3.81·18-s − 0.268·19-s + 1.82·21-s + 0.296·22-s − 0.00537·23-s − 6.72·24-s − 0.872·26-s + 1.55·27-s + 3.18·28-s − 0.829·29-s − 1.15·31-s − 5.79·32-s − 0.254·33-s − 1.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.011341490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011341490\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 2 | \( 1 + 2.82T + 2T^{2} \) |
| 3 | \( 1 - 2.95T + 3T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 0.493T + 11T^{2} \) |
| 13 | \( 1 - 1.57T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.0257T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 - 5.77T + 37T^{2} \) |
| 41 | \( 1 - 5.69T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 - 7.51T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 + 1.95T + 71T^{2} \) |
| 73 | \( 1 + 0.678T + 73T^{2} \) |
| 79 | \( 1 - 9.30T + 79T^{2} \) |
| 83 | \( 1 + 2.38T + 83T^{2} \) |
| 89 | \( 1 + 0.754T + 89T^{2} \) |
| 97 | \( 1 + 0.821T + 97T^{2} \) |
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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
−8.146160656592436842570708696381, −7.60315971555218224886640864708, −7.37876809031947193075095579087, −6.30995432873405630337382652142, −5.41780986466498470122611975901, −3.97383384926688126600728397242, −3.20852248356716670041076132697, −2.33382294145197173107679238613, −1.81056196248374939397957293880, −0.959653734271085258979497374023,
0.959653734271085258979497374023, 1.81056196248374939397957293880, 2.33382294145197173107679238613, 3.20852248356716670041076132697, 3.97383384926688126600728397242, 5.41780986466498470122611975901, 6.30995432873405630337382652142, 7.37876809031947193075095579087, 7.60315971555218224886640864708, 8.146160656592436842570708696381
