| L(s) = 1 | + 2-s + 4-s − 3.97·5-s + 3.97·7-s + 8-s − 3.97·10-s + 4.91·11-s + 6.34·13-s + 3.97·14-s + 16-s + 3.91·17-s + 7.78·19-s − 3.97·20-s + 4.91·22-s + 0.326·23-s + 10.8·25-s + 6.34·26-s + 3.97·28-s − 2.16·29-s + 3.23·31-s + 32-s + 3.91·34-s − 15.8·35-s − 3.13·37-s + 7.78·38-s − 3.97·40-s − 2.94·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.77·5-s + 1.50·7-s + 0.353·8-s − 1.25·10-s + 1.48·11-s + 1.76·13-s + 1.06·14-s + 0.250·16-s + 0.949·17-s + 1.78·19-s − 0.889·20-s + 1.04·22-s + 0.0681·23-s + 2.16·25-s + 1.24·26-s + 0.751·28-s − 0.402·29-s + 0.581·31-s + 0.176·32-s + 0.671·34-s − 2.67·35-s − 0.516·37-s + 1.26·38-s − 0.628·40-s − 0.460·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.073620324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.073620324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 19 | \( 1 - 7.78T + 19T^{2} \) |
| 23 | \( 1 - 0.326T + 23T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 + 8.02T + 43T^{2} \) |
| 47 | \( 1 + 0.363T + 47T^{2} \) |
| 53 | \( 1 + 8.28T + 53T^{2} \) |
| 59 | \( 1 - 1.35T + 59T^{2} \) |
| 61 | \( 1 - 2.36T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.925986590006279141268192051862, −7.13834027111152975770310765329, −6.55716224414393440934282303871, −5.52427904375238393106454027545, −4.94633326726191877588587378447, −4.11601372640059062782069810471, −3.66774393958073837302842660679, −3.16688499687454972671070293009, −1.44054101046361020883622029952, −1.10632971087460857890047269536,
1.10632971087460857890047269536, 1.44054101046361020883622029952, 3.16688499687454972671070293009, 3.66774393958073837302842660679, 4.11601372640059062782069810471, 4.94633326726191877588587378447, 5.52427904375238393106454027545, 6.55716224414393440934282303871, 7.13834027111152975770310765329, 7.925986590006279141268192051862
