| L(s) = 1 | + 1.72·3-s − 0.341·5-s + 4.71·7-s − 0.0113·9-s + 4.94·11-s − 5.88·13-s − 0.590·15-s + 1.39·17-s − 2.53·19-s + 8.15·21-s + 4.40·23-s − 4.88·25-s − 5.20·27-s − 3.15·29-s − 5.25·31-s + 8.54·33-s − 1.61·35-s + 11.6·37-s − 10.1·39-s + 9.59·41-s − 3.73·43-s + 0.00388·45-s + 9.83·47-s + 15.2·49-s + 2.41·51-s + 10.2·53-s − 1.68·55-s + ⋯ |
| L(s) = 1 | + 0.998·3-s − 0.152·5-s + 1.78·7-s − 0.00379·9-s + 1.49·11-s − 1.63·13-s − 0.152·15-s + 0.339·17-s − 0.580·19-s + 1.77·21-s + 0.917·23-s − 0.976·25-s − 1.00·27-s − 0.585·29-s − 0.944·31-s + 1.48·33-s − 0.272·35-s + 1.92·37-s − 1.62·39-s + 1.49·41-s − 0.569·43-s + 0.000579·45-s + 1.43·47-s + 2.17·49-s + 0.338·51-s + 1.40·53-s − 0.227·55-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
| \(L(1)\) |
\(\approx\) |
\(3.575124820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.575124820\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.72T + 3T^{2} \) |
| 5 | \( 1 + 0.341T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 + 5.88T + 13T^{2} \) |
| 17 | \( 1 - 1.39T + 17T^{2} \) |
| 19 | \( 1 + 2.53T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 3.15T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 - 9.59T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 7.34T + 61T^{2} \) |
| 67 | \( 1 + 0.205T + 67T^{2} \) |
| 71 | \( 1 + 5.31T + 71T^{2} \) |
| 73 | \( 1 - 6.10T + 73T^{2} \) |
| 79 | \( 1 - 4.27T + 79T^{2} \) |
| 83 | \( 1 - 7.38T + 83T^{2} \) |
| 89 | \( 1 - 1.88T + 89T^{2} \) |
| 97 | \( 1 - 6.65T + 97T^{2} \) |
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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
−7.994397898635328192389442281193, −7.57210425051143693788390420170, −6.99630755007188374398355454129, −5.79497946483791315329889195514, −5.15418197123323594672838914833, −4.19748580088109832650509505214, −3.87028771852566506071306905184, −2.49174222670860983660070926744, −2.10340653438424042001885764274, −0.988584717473526133859467233057,
0.988584717473526133859467233057, 2.10340653438424042001885764274, 2.49174222670860983660070926744, 3.87028771852566506071306905184, 4.19748580088109832650509505214, 5.15418197123323594672838914833, 5.79497946483791315329889195514, 6.99630755007188374398355454129, 7.57210425051143693788390420170, 7.994397898635328192389442281193
