| L(s) = 1 | − 2-s − 2.64·3-s + 4-s + 2.64·6-s − 8-s + 4.00·9-s + 5·11-s − 2.64·12-s + 5.29·13-s + 16-s − 2.64·17-s − 4.00·18-s + 7.93·19-s − 5·22-s + 4·23-s + 2.64·24-s − 5.29·26-s − 2.64·27-s + 6·29-s − 10.5·31-s − 32-s − 13.2·33-s + 2.64·34-s + 4.00·36-s + 4·37-s − 7.93·38-s − 14.0·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.52·3-s + 0.5·4-s + 1.08·6-s − 0.353·8-s + 1.33·9-s + 1.50·11-s − 0.763·12-s + 1.46·13-s + 0.250·16-s − 0.641·17-s − 0.942·18-s + 1.82·19-s − 1.06·22-s + 0.834·23-s + 0.540·24-s − 1.03·26-s − 0.509·27-s + 1.11·29-s − 1.90·31-s − 0.176·32-s − 2.30·33-s + 0.453·34-s + 0.666·36-s + 0.657·37-s − 1.28·38-s − 2.24·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9439152644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9439152644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 11 | \( 1 - 5T + 11T^{2} \) |
| 13 | \( 1 - 5.29T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 7.93T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10.5T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 5.29T + 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 2.64T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.195460609224973651153815771078, −8.235465587988382025082081230833, −7.13592978412760142376601780330, −6.64255671187350039413442367943, −5.97108860711102937974502999916, −5.26534483171923288050751405476, −4.19667392295124004909934217038, −3.25510390958832080300377806416, −1.48560020886976298096683160148, −0.846757410719185796178092076935,
0.846757410719185796178092076935, 1.48560020886976298096683160148, 3.25510390958832080300377806416, 4.19667392295124004909934217038, 5.26534483171923288050751405476, 5.97108860711102937974502999916, 6.64255671187350039413442367943, 7.13592978412760142376601780330, 8.235465587988382025082081230833, 9.195460609224973651153815771078
