| L(s) = 1 | − 2.83·3-s + 4.26·5-s + 3.89·7-s + 5.01·9-s + 2.11·11-s + 2.93·13-s − 12.0·15-s − 2.84·17-s − 7.25·19-s − 11.0·21-s + 0.341·23-s + 13.1·25-s − 5.71·27-s + 0.780·29-s + 5.53·31-s − 6.00·33-s + 16.6·35-s + 6.82·37-s − 8.31·39-s − 8.86·41-s − 4.55·43-s + 21.3·45-s − 9.00·47-s + 8.20·49-s + 8.05·51-s + 4.97·53-s + 9.03·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 1.90·5-s + 1.47·7-s + 1.67·9-s + 0.638·11-s + 0.814·13-s − 3.11·15-s − 0.689·17-s − 1.66·19-s − 2.40·21-s + 0.0712·23-s + 2.63·25-s − 1.09·27-s + 0.144·29-s + 0.994·31-s − 1.04·33-s + 2.81·35-s + 1.12·37-s − 1.33·39-s − 1.38·41-s − 0.694·43-s + 3.18·45-s − 1.31·47-s + 1.17·49-s + 1.12·51-s + 0.683·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258019599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258019599\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 - 4.26T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 13 | \( 1 - 2.93T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 0.341T + 23T^{2} \) |
| 29 | \( 1 - 0.780T + 29T^{2} \) |
| 31 | \( 1 - 5.53T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 + 8.86T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 - 1.52T + 59T^{2} \) |
| 61 | \( 1 - 7.97T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 8.40T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 - 4.05T + 97T^{2} \) |
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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
−8.306301332359346106466976483767, −6.87149125625021283854116961875, −6.38783089132247508741484254217, −6.11456658779380964434113829641, −5.07601460527558048573991258101, −4.94853796411318027220087265175, −4.01154838104988148761751028789, −2.30620647605242081279930081730, −1.67756302723900799830886467230, −0.941620008776457648681087815289,
0.941620008776457648681087815289, 1.67756302723900799830886467230, 2.30620647605242081279930081730, 4.01154838104988148761751028789, 4.94853796411318027220087265175, 5.07601460527558048573991258101, 6.11456658779380964434113829641, 6.38783089132247508741484254217, 6.87149125625021283854116961875, 8.306301332359346106466976483767
