| L(s) = 1 | − 2-s − 0.414·3-s + 4-s + 5-s + 0.414·6-s − 0.282·7-s − 8-s − 2.82·9-s − 10-s − 5.27·11-s − 0.414·12-s + 1.86·13-s + 0.282·14-s − 0.414·15-s + 16-s + 5.24·17-s + 2.82·18-s + 0.946·19-s + 20-s + 0.117·21-s + 5.27·22-s + 0.414·24-s + 25-s − 1.86·26-s + 2.41·27-s − 0.282·28-s − 1.64·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s − 0.106·7-s − 0.353·8-s − 0.942·9-s − 0.316·10-s − 1.59·11-s − 0.119·12-s + 0.516·13-s + 0.0755·14-s − 0.106·15-s + 0.250·16-s + 1.27·17-s + 0.666·18-s + 0.217·19-s + 0.223·20-s + 0.0255·21-s + 1.12·22-s + 0.0845·24-s + 0.200·25-s − 0.365·26-s + 0.464·27-s − 0.0533·28-s − 0.305·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9529841821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9529841821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 7 | \( 1 + 0.282T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 5.24T + 17T^{2} \) |
| 19 | \( 1 - 0.946T + 19T^{2} \) |
| 29 | \( 1 + 1.64T + 29T^{2} \) |
| 31 | \( 1 - 3.85T + 31T^{2} \) |
| 37 | \( 1 + 0.828T + 37T^{2} \) |
| 41 | \( 1 + 7.53T + 41T^{2} \) |
| 43 | \( 1 - 0.663T + 43T^{2} \) |
| 47 | \( 1 - 0.353T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 9.75T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 - 4.81T + 79T^{2} \) |
| 83 | \( 1 + 0.643T + 83T^{2} \) |
| 89 | \( 1 - 2.29T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.215948910807269807444551988184, −7.66379919296517055324761073466, −6.83885958844008260742972151086, −5.85228993540137151510092464483, −5.60664083992839375463139554010, −4.75364368387109428751573788756, −3.30192209189091941858891456081, −2.84136043159242683649152167248, −1.78916797815019317216156552801, −0.58104705264863304550105568500,
0.58104705264863304550105568500, 1.78916797815019317216156552801, 2.84136043159242683649152167248, 3.30192209189091941858891456081, 4.75364368387109428751573788756, 5.60664083992839375463139554010, 5.85228993540137151510092464483, 6.83885958844008260742972151086, 7.66379919296517055324761073466, 8.215948910807269807444551988184
