| L(s) = 1 | + 2.48·3-s − 1.91·5-s − 3.48·7-s + 3.17·9-s + 11-s − 1.66·13-s − 4.74·15-s − 1.94·17-s + 19-s − 8.65·21-s − 5.70·23-s − 1.35·25-s + 0.425·27-s − 1.22·29-s − 0.402·31-s + 2.48·33-s + 6.65·35-s − 9.65·37-s − 4.13·39-s + 8.40·41-s + 0.648·43-s − 6.05·45-s − 5.64·47-s + 5.13·49-s − 4.84·51-s − 2.17·53-s − 1.91·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 0.854·5-s − 1.31·7-s + 1.05·9-s + 0.301·11-s − 0.461·13-s − 1.22·15-s − 0.472·17-s + 0.229·19-s − 1.88·21-s − 1.19·23-s − 0.270·25-s + 0.0819·27-s − 0.227·29-s − 0.0722·31-s + 0.432·33-s + 1.12·35-s − 1.58·37-s − 0.661·39-s + 1.31·41-s + 0.0989·43-s − 0.903·45-s − 0.822·47-s + 0.734·49-s − 0.677·51-s − 0.299·53-s − 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 7 | \( 1 + 3.48T + 7T^{2} \) |
| 13 | \( 1 + 1.66T + 13T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 + 0.402T + 31T^{2} \) |
| 37 | \( 1 + 9.65T + 37T^{2} \) |
| 41 | \( 1 - 8.40T + 41T^{2} \) |
| 43 | \( 1 - 0.648T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 6.29T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 + 0.672T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 + 7.20T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−8.974116226303335432746362808439, −8.190341682202155215118191923669, −7.51831514042683205983290609782, −6.81195553704580019159634641423, −5.83407278707771307186869884156, −4.38515826422938212282941004148, −3.65761912716795085031796606495, −3.05155868206136555865478313723, −2.00958883567972918932124564704, 0,
2.00958883567972918932124564704, 3.05155868206136555865478313723, 3.65761912716795085031796606495, 4.38515826422938212282941004148, 5.83407278707771307186869884156, 6.81195553704580019159634641423, 7.51831514042683205983290609782, 8.190341682202155215118191923669, 8.974116226303335432746362808439
