| L(s) = 1 | + 2.64·2-s + 4.97·4-s − 5-s + 3.18·7-s + 7.86·8-s − 2.64·10-s + 1.72·11-s − 0.130·13-s + 8.42·14-s + 10.8·16-s − 0.163·17-s − 3.09·19-s − 4.97·20-s + 4.54·22-s − 0.0695·23-s + 25-s − 0.344·26-s + 15.8·28-s + 1.18·29-s + 2.71·31-s + 12.8·32-s − 0.431·34-s − 3.18·35-s + 10.6·37-s − 8.17·38-s − 7.86·40-s + 0.216·41-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 2.48·4-s − 0.447·5-s + 1.20·7-s + 2.78·8-s − 0.835·10-s + 0.518·11-s − 0.0361·13-s + 2.25·14-s + 2.70·16-s − 0.0396·17-s − 0.709·19-s − 1.11·20-s + 0.969·22-s − 0.0145·23-s + 0.200·25-s − 0.0674·26-s + 2.99·28-s + 0.220·29-s + 0.488·31-s + 2.27·32-s − 0.0740·34-s − 0.538·35-s + 1.75·37-s − 1.32·38-s − 1.24·40-s + 0.0337·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.571938657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.571938657\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 107 | \( 1 + T \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 1.72T + 11T^{2} \) |
| 13 | \( 1 + 0.130T + 13T^{2} \) |
| 17 | \( 1 + 0.163T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 + 0.0695T + 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 0.216T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 + 7.17T + 47T^{2} \) |
| 53 | \( 1 + 5.50T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 0.304T + 61T^{2} \) |
| 67 | \( 1 + 6.06T + 67T^{2} \) |
| 71 | \( 1 - 4.92T + 71T^{2} \) |
| 73 | \( 1 - 0.0247T + 73T^{2} \) |
| 79 | \( 1 - 4.89T + 79T^{2} \) |
| 83 | \( 1 + 6.00T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 17.6T + 97T^{2} \) |
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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
−7.919425806099897634725668324255, −7.45827016503712066506611638290, −6.50555156160023714608872799691, −6.02690876493329031072308381085, −5.09469994361490136698710310383, −4.47384614922697227042269329662, −4.09468677172906073298791831113, −3.08110273068640594405112216005, −2.24775538920208487276139467786, −1.29474538267094646636577280493,
1.29474538267094646636577280493, 2.24775538920208487276139467786, 3.08110273068640594405112216005, 4.09468677172906073298791831113, 4.47384614922697227042269329662, 5.09469994361490136698710310383, 6.02690876493329031072308381085, 6.50555156160023714608872799691, 7.45827016503712066506611638290, 7.919425806099897634725668324255
