| L(s) = 1 | − 0.0391·3-s + 3.50·5-s − 3.34·7-s − 2.99·9-s + 2.61·11-s − 0.169·13-s − 0.137·15-s + 17-s − 1.54·19-s + 0.131·21-s + 4.43·23-s + 7.27·25-s + 0.234·27-s − 3.94·29-s + 0.0167·31-s − 0.102·33-s − 11.7·35-s − 6.89·37-s + 0.00663·39-s − 5.91·41-s − 5.05·43-s − 10.5·45-s − 10.2·47-s + 4.20·49-s − 0.0391·51-s + 4.84·53-s + 9.14·55-s + ⋯ |
| L(s) = 1 | − 0.0226·3-s + 1.56·5-s − 1.26·7-s − 0.999·9-s + 0.787·11-s − 0.0469·13-s − 0.0354·15-s + 0.242·17-s − 0.353·19-s + 0.0286·21-s + 0.923·23-s + 1.45·25-s + 0.0452·27-s − 0.732·29-s + 0.00299·31-s − 0.0178·33-s − 1.98·35-s − 1.13·37-s + 0.00106·39-s − 0.923·41-s − 0.770·43-s − 1.56·45-s − 1.49·47-s + 0.600·49-s − 0.00548·51-s + 0.666·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0391T + 3T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 - 2.61T + 11T^{2} \) |
| 13 | \( 1 + 0.169T + 13T^{2} \) |
| 19 | \( 1 + 1.54T + 19T^{2} \) |
| 23 | \( 1 - 4.43T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 0.0167T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 + 5.91T + 41T^{2} \) |
| 43 | \( 1 + 5.05T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 61 | \( 1 - 5.53T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 4.46T + 71T^{2} \) |
| 73 | \( 1 + 2.81T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 - 4.60T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.24844936287221665093098031771, −6.49466399569570435185765378700, −6.27226471350157790504021241590, −5.48517703448994054180567026466, −4.94923399080842882546946600829, −3.61575452490499677013573808611, −3.10863831081514464137542586910, −2.25152351854249144676867935318, −1.38198662376251526972430609639, 0,
1.38198662376251526972430609639, 2.25152351854249144676867935318, 3.10863831081514464137542586910, 3.61575452490499677013573808611, 4.94923399080842882546946600829, 5.48517703448994054180567026466, 6.27226471350157790504021241590, 6.49466399569570435185765378700, 7.24844936287221665093098031771
