| L(s) = 1 | − 1.28·2-s − 0.361·4-s + 4.05·5-s − 2.17·7-s + 3.02·8-s − 5.19·10-s − 3.17·11-s + 2.18·13-s + 2.77·14-s − 3.14·16-s − 2.58·17-s + 19-s − 1.46·20-s + 4.06·22-s + 7.11·23-s + 11.4·25-s − 2.79·26-s + 0.784·28-s + 7.08·29-s + 4.78·31-s − 2.01·32-s + 3.31·34-s − 8.81·35-s − 5.35·37-s − 1.28·38-s + 12.2·40-s − 2.87·41-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.180·4-s + 1.81·5-s − 0.820·7-s + 1.06·8-s − 1.64·10-s − 0.957·11-s + 0.605·13-s + 0.742·14-s − 0.786·16-s − 0.627·17-s + 0.229·19-s − 0.328·20-s + 0.866·22-s + 1.48·23-s + 2.29·25-s − 0.547·26-s + 0.148·28-s + 1.31·29-s + 0.859·31-s − 0.356·32-s + 0.568·34-s − 1.48·35-s − 0.880·37-s − 0.207·38-s + 1.94·40-s − 0.448·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.442697772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442697772\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 + 1.28T + 2T^{2} \) |
| 5 | \( 1 - 4.05T + 5T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 - 7.08T + 29T^{2} \) |
| 31 | \( 1 - 4.78T + 31T^{2} \) |
| 37 | \( 1 + 5.35T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 - 5.09T + 43T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 - 0.245T + 61T^{2} \) |
| 67 | \( 1 - 7.94T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 + 1.67T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 9.71T + 97T^{2} \) |
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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
−8.107154881795499253423651322019, −6.92633632258823820966900754259, −6.68457398073607148944393098717, −5.77320138714262899426945980923, −5.15727620570100525207983891837, −4.50870327890571217050635425101, −3.16710697441796023393612232887, −2.54048744038231644664908885667, −1.58357082964694996257866037709, −0.71703750865604334188385655009,
0.71703750865604334188385655009, 1.58357082964694996257866037709, 2.54048744038231644664908885667, 3.16710697441796023393612232887, 4.50870327890571217050635425101, 5.15727620570100525207983891837, 5.77320138714262899426945980923, 6.68457398073607148944393098717, 6.92633632258823820966900754259, 8.107154881795499253423651322019
