| L(s) = 1 | − 2·2-s + 10·3-s + 4·4-s − 20·6-s + 7·7-s − 8·8-s + 73·9-s + 9·11-s + 40·12-s − 52·13-s − 14·14-s + 16·16-s + 96·17-s − 146·18-s − 10·19-s + 70·21-s − 18·22-s + 75·23-s − 80·24-s + 104·26-s + 460·27-s + 28·28-s + 189·29-s − 232·31-s − 32·32-s + 90·33-s − 192·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.92·3-s + 1/2·4-s − 1.36·6-s + 0.377·7-s − 0.353·8-s + 2.70·9-s + 0.246·11-s + 0.962·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.36·17-s − 1.91·18-s − 0.120·19-s + 0.727·21-s − 0.174·22-s + 0.679·23-s − 0.680·24-s + 0.784·26-s + 3.27·27-s + 0.188·28-s + 1.21·29-s − 1.34·31-s − 0.176·32-s + 0.474·33-s − 0.968·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.949626749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.949626749\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 10 T + p^{3} T^{2} \) |
| 23 | \( 1 - 75 T + p^{3} T^{2} \) |
| 29 | \( 1 - 189 T + p^{3} T^{2} \) |
| 31 | \( 1 + 232 T + p^{3} T^{2} \) |
| 37 | \( 1 - 305 T + p^{3} T^{2} \) |
| 41 | \( 1 + 438 T + p^{3} T^{2} \) |
| 43 | \( 1 - 353 T + p^{3} T^{2} \) |
| 47 | \( 1 + 486 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 + 672 T + p^{3} T^{2} \) |
| 61 | \( 1 - 206 T + p^{3} T^{2} \) |
| 67 | \( 1 - 599 T + p^{3} T^{2} \) |
| 71 | \( 1 + 471 T + p^{3} T^{2} \) |
| 73 | \( 1 - 614 T + p^{3} T^{2} \) |
| 79 | \( 1 - 743 T + p^{3} T^{2} \) |
| 83 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 180 T + p^{3} T^{2} \) |
| 97 | \( 1 + 184 T + p^{3} T^{2} \) |
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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
−10.67578773904240973083378679362, −9.705693592679979661925070429855, −9.277967185900813285411063317474, −8.164895386292406078619635083769, −7.72407337362098831928992378282, −6.78031007921562599657583235141, −4.88311035495610326508509429800, −3.49389185098859385609006115477, −2.53173862327978090215295081586, −1.35664056388397445209219995392,
1.35664056388397445209219995392, 2.53173862327978090215295081586, 3.49389185098859385609006115477, 4.88311035495610326508509429800, 6.78031007921562599657583235141, 7.72407337362098831928992378282, 8.164895386292406078619635083769, 9.277967185900813285411063317474, 9.705693592679979661925070429855, 10.67578773904240973083378679362
