| L(s) = 1 | − 2.49·3-s + 2.49·7-s + 3.20·9-s − 3.49·11-s − 0.713·13-s + 3.91·17-s − 19-s − 6.20·21-s − 2.20·23-s − 0.509·27-s + 0.636·29-s − 2.14·31-s + 8.69·33-s + 2.06·37-s + 1.77·39-s − 4.35·41-s − 4.55·43-s + 0.268·47-s − 0.795·49-s − 9.75·51-s + 7.98·53-s + 2.49·57-s − 2.57·59-s − 10.8·61-s + 7.98·63-s − 1.08·67-s + 5.49·69-s + ⋯ |
| L(s) = 1 | − 1.43·3-s + 0.941·7-s + 1.06·9-s − 1.05·11-s − 0.197·13-s + 0.950·17-s − 0.229·19-s − 1.35·21-s − 0.459·23-s − 0.0979·27-s + 0.118·29-s − 0.384·31-s + 1.51·33-s + 0.339·37-s + 0.284·39-s − 0.679·41-s − 0.694·43-s + 0.0391·47-s − 0.113·49-s − 1.36·51-s + 1.09·53-s + 0.329·57-s − 0.334·59-s − 1.38·61-s + 1.00·63-s − 0.132·67-s + 0.661·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.713T + 13T^{2} \) |
| 17 | \( 1 - 3.91T + 17T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 0.636T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 + 4.35T + 41T^{2} \) |
| 43 | \( 1 + 4.55T + 43T^{2} \) |
| 47 | \( 1 - 0.268T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.08T + 67T^{2} \) |
| 71 | \( 1 - 6.83T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 7.63T + 79T^{2} \) |
| 83 | \( 1 - 0.923T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−7.943036602010767857566676303212, −7.45271506710809212364603004843, −6.48308885865661534165632782739, −5.77927208211614934619193280526, −5.09779379196637661083727624603, −4.73895646290322772409257666376, −3.56768623641169398570678218423, −2.32137982425171124118785129305, −1.21396617733099083541986433017, 0,
1.21396617733099083541986433017, 2.32137982425171124118785129305, 3.56768623641169398570678218423, 4.73895646290322772409257666376, 5.09779379196637661083727624603, 5.77927208211614934619193280526, 6.48308885865661534165632782739, 7.45271506710809212364603004843, 7.943036602010767857566676303212
