| L(s) = 1 | + 0.431·2-s − 1.81·4-s + 3.65·5-s + 2.43·7-s − 1.64·8-s + 1.57·10-s + 6.02·11-s − 5.27·13-s + 1.04·14-s + 2.91·16-s + 6.46·17-s + 0.985·19-s − 6.62·20-s + 2.60·22-s + 23-s + 8.33·25-s − 2.27·26-s − 4.40·28-s + 29-s + 9.09·31-s + 4.54·32-s + 2.78·34-s + 8.87·35-s − 9.30·37-s + 0.425·38-s − 6.00·40-s − 8.21·41-s + ⋯ |
| L(s) = 1 | + 0.305·2-s − 0.906·4-s + 1.63·5-s + 0.918·7-s − 0.581·8-s + 0.498·10-s + 1.81·11-s − 1.46·13-s + 0.280·14-s + 0.729·16-s + 1.56·17-s + 0.226·19-s − 1.48·20-s + 0.554·22-s + 0.208·23-s + 1.66·25-s − 0.445·26-s − 0.833·28-s + 0.185·29-s + 1.63·31-s + 0.804·32-s + 0.478·34-s + 1.50·35-s − 1.52·37-s + 0.0689·38-s − 0.949·40-s − 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.401598617\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.401598617\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.431T + 2T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 - 6.46T + 17T^{2} \) |
| 19 | \( 1 - 0.985T + 19T^{2} \) |
| 31 | \( 1 - 9.09T + 31T^{2} \) |
| 37 | \( 1 + 9.30T + 37T^{2} \) |
| 41 | \( 1 + 8.21T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 - 7.09T + 53T^{2} \) |
| 59 | \( 1 + 2.46T + 59T^{2} \) |
| 61 | \( 1 - 1.46T + 61T^{2} \) |
| 67 | \( 1 + 4.39T + 67T^{2} \) |
| 71 | \( 1 - 5.24T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 2.77T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.262111939277646817988510062533, −7.27063291038028173361205731465, −6.53058426786938927790534284785, −5.76301328864945704637537057843, −5.13675337073755169672099429698, −4.71696115190260019186689933057, −3.71875375500909470401395387977, −2.79749874745252604063271564576, −1.69379636673241410796616831748, −1.05096651905004697980083523258,
1.05096651905004697980083523258, 1.69379636673241410796616831748, 2.79749874745252604063271564576, 3.71875375500909470401395387977, 4.71696115190260019186689933057, 5.13675337073755169672099429698, 5.76301328864945704637537057843, 6.53058426786938927790534284785, 7.27063291038028173361205731465, 8.262111939277646817988510062533
