| L(s) = 1 | + 2.74·2-s + 5.55·4-s + 5-s + 4.26·7-s + 9.77·8-s + 2.74·10-s − 0.899·11-s − 6.86·13-s + 11.7·14-s + 15.7·16-s − 2.01·17-s + 8.38·19-s + 5.55·20-s − 2.47·22-s − 6.70·23-s + 25-s − 18.8·26-s + 23.6·28-s + 2.96·29-s − 5.52·31-s + 23.7·32-s − 5.53·34-s + 4.26·35-s − 5.73·37-s + 23.0·38-s + 9.77·40-s − 1.02·41-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 2.77·4-s + 0.447·5-s + 1.61·7-s + 3.45·8-s + 0.869·10-s − 0.271·11-s − 1.90·13-s + 3.13·14-s + 3.93·16-s − 0.488·17-s + 1.92·19-s + 1.24·20-s − 0.527·22-s − 1.39·23-s + 0.200·25-s − 3.70·26-s + 4.47·28-s + 0.550·29-s − 0.992·31-s + 4.20·32-s − 0.949·34-s + 0.720·35-s − 0.942·37-s + 3.73·38-s + 1.54·40-s − 0.160·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.729818423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.729818423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 11 | \( 1 + 0.899T + 11T^{2} \) |
| 13 | \( 1 + 6.86T + 13T^{2} \) |
| 17 | \( 1 + 2.01T + 17T^{2} \) |
| 19 | \( 1 - 8.38T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2.96T + 29T^{2} \) |
| 31 | \( 1 + 5.52T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 - 4.46T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 4.95T + 73T^{2} \) |
| 79 | \( 1 + 7.52T + 79T^{2} \) |
| 83 | \( 1 + 0.785T + 83T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−7.909880464799646660563245948728, −7.50769780192558801554816246762, −6.94305288817015477624487042844, −5.71055219171066291933728526649, −5.39150426499884895910182064542, −4.72853286658482322926321923464, −4.17902830414718182657082389114, −2.98018912091017767923425660707, −2.25883769105341598033422547112, −1.55023685782082488075135993024,
1.55023685782082488075135993024, 2.25883769105341598033422547112, 2.98018912091017767923425660707, 4.17902830414718182657082389114, 4.72853286658482322926321923464, 5.39150426499884895910182064542, 5.71055219171066291933728526649, 6.94305288817015477624487042844, 7.50769780192558801554816246762, 7.909880464799646660563245948728
