| L(s) = 1 | − 3.13·3-s + 0.831·5-s + 6.83·9-s − 1.98·11-s − 3.37·13-s − 2.60·15-s − 0.162·17-s + 1.67·19-s + 23-s − 4.30·25-s − 12.0·27-s + 7.28·29-s − 1.38·31-s + 6.21·33-s + 8.72·37-s + 10.5·39-s − 5.38·41-s + 2.93·43-s + 5.68·45-s − 0.950·47-s + 0.510·51-s − 5.52·53-s − 1.64·55-s − 5.24·57-s − 7.29·59-s + 5.39·61-s − 2.80·65-s + ⋯ |
| L(s) = 1 | − 1.81·3-s + 0.371·5-s + 2.27·9-s − 0.597·11-s − 0.935·13-s − 0.673·15-s − 0.0395·17-s + 0.383·19-s + 0.208·23-s − 0.861·25-s − 2.31·27-s + 1.35·29-s − 0.248·31-s + 1.08·33-s + 1.43·37-s + 1.69·39-s − 0.840·41-s + 0.447·43-s + 0.846·45-s − 0.138·47-s + 0.0715·51-s − 0.758·53-s − 0.222·55-s − 0.694·57-s − 0.949·59-s + 0.690·61-s − 0.347·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 - 0.831T + 5T^{2} \) |
| 11 | \( 1 + 1.98T + 11T^{2} \) |
| 13 | \( 1 + 3.37T + 13T^{2} \) |
| 17 | \( 1 + 0.162T + 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 8.72T + 37T^{2} \) |
| 41 | \( 1 + 5.38T + 41T^{2} \) |
| 43 | \( 1 - 2.93T + 43T^{2} \) |
| 47 | \( 1 + 0.950T + 47T^{2} \) |
| 53 | \( 1 + 5.52T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 4.43T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 - 7.13T + 97T^{2} \) |
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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
−7.33096509771093242308338113850, −6.46991559060516405938255486643, −6.03997377589191212786326407955, −5.31858267491286842919513038052, −4.83563020835468930985591169888, −4.22476908443434335997496211630, −2.99785087891733384989157296739, −2.00616809645224003936617261156, −0.968460222482656174556740418504, 0,
0.968460222482656174556740418504, 2.00616809645224003936617261156, 2.99785087891733384989157296739, 4.22476908443434335997496211630, 4.83563020835468930985591169888, 5.31858267491286842919513038052, 6.03997377589191212786326407955, 6.46991559060516405938255486643, 7.33096509771093242308338113850
