| L(s) = 1 | + 3.29·3-s − 0.255·5-s + 7.83·9-s + 5.29·11-s + 5.00·13-s − 0.841·15-s − 17-s + 4.78·19-s − 2.20·23-s − 4.93·25-s + 15.9·27-s − 9.21·29-s − 10.4·31-s + 17.4·33-s + 2.45·37-s + 16.4·39-s − 3.70·41-s + 9.09·43-s − 2.00·45-s − 3.08·47-s − 3.29·51-s − 3.01·53-s − 1.35·55-s + 15.7·57-s − 8.02·59-s − 1.32·61-s − 1.27·65-s + ⋯ |
| L(s) = 1 | + 1.90·3-s − 0.114·5-s + 2.61·9-s + 1.59·11-s + 1.38·13-s − 0.217·15-s − 0.242·17-s + 1.09·19-s − 0.459·23-s − 0.986·25-s + 3.06·27-s − 1.71·29-s − 1.87·31-s + 3.03·33-s + 0.403·37-s + 2.63·39-s − 0.578·41-s + 1.38·43-s − 0.298·45-s − 0.450·47-s − 0.460·51-s − 0.413·53-s − 0.182·55-s + 2.08·57-s − 1.04·59-s − 0.169·61-s − 0.158·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.485737660\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.485737660\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 5 | \( 1 + 0.255T + 5T^{2} \) |
| 11 | \( 1 - 5.29T + 11T^{2} \) |
| 13 | \( 1 - 5.00T + 13T^{2} \) |
| 19 | \( 1 - 4.78T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 9.09T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 + 8.02T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 2.39T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 3.14T + 79T^{2} \) |
| 83 | \( 1 - 7.79T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.855030857430132089092454736847, −7.81677435096291897239955346134, −7.49991785942011047067275519888, −6.56451082728508960424300905688, −5.70935633126933544826944550827, −4.27072067636669136190551254019, −3.71879737786751345914468132891, −3.31074229452828336802700565226, −1.95318806206666422542220214243, −1.38817457350297941992360574867,
1.38817457350297941992360574867, 1.95318806206666422542220214243, 3.31074229452828336802700565226, 3.71879737786751345914468132891, 4.27072067636669136190551254019, 5.70935633126933544826944550827, 6.56451082728508960424300905688, 7.49991785942011047067275519888, 7.81677435096291897239955346134, 8.855030857430132089092454736847
