| L(s) = 1 | + 2·2-s + 4·4-s + 20·5-s − 7·7-s + 8·8-s + 40·10-s + 28·11-s + 3·13-s − 14·14-s + 16·16-s + 115·17-s − 106·19-s + 80·20-s + 56·22-s − 149·23-s + 275·25-s + 6·26-s − 28·28-s + 11·29-s + 81·31-s + 32·32-s + 230·34-s − 140·35-s + 2·37-s − 212·38-s + 160·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s + 0.353·8-s + 1.26·10-s + 0.767·11-s + 0.0640·13-s − 0.267·14-s + 1/4·16-s + 1.64·17-s − 1.27·19-s + 0.894·20-s + 0.542·22-s − 1.35·23-s + 11/5·25-s + 0.0452·26-s − 0.188·28-s + 0.0704·29-s + 0.469·31-s + 0.176·32-s + 1.16·34-s − 0.676·35-s + 0.00888·37-s − 0.905·38-s + 0.632·40-s − 0.0228·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.105825351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.105825351\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 11 | \( 1 - 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 3 T + p^{3} T^{2} \) |
| 17 | \( 1 - 115 T + p^{3} T^{2} \) |
| 19 | \( 1 + 106 T + p^{3} T^{2} \) |
| 23 | \( 1 + 149 T + p^{3} T^{2} \) |
| 29 | \( 1 - 11 T + p^{3} T^{2} \) |
| 31 | \( 1 - 81 T + p^{3} T^{2} \) |
| 37 | \( 1 - 2 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 139 T + p^{3} T^{2} \) |
| 47 | \( 1 - 474 T + p^{3} T^{2} \) |
| 53 | \( 1 + 531 T + p^{3} T^{2} \) |
| 59 | \( 1 + 817 T + p^{3} T^{2} \) |
| 61 | \( 1 - 498 T + p^{3} T^{2} \) |
| 67 | \( 1 - 793 T + p^{3} T^{2} \) |
| 71 | \( 1 - 853 T + p^{3} T^{2} \) |
| 73 | \( 1 - 490 T + p^{3} T^{2} \) |
| 79 | \( 1 + 330 T + p^{3} T^{2} \) |
| 83 | \( 1 + 404 T + p^{3} T^{2} \) |
| 89 | \( 1 - 831 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1424 T + p^{3} T^{2} \) |
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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
−10.82395943167387627638475129219, −9.998102159020261865955060941052, −9.410920192079793164617988334720, −8.138393146796464789814199462339, −6.64134267177989658435233209515, −6.09819524296858142346616504686, −5.26887457404686642434919460938, −3.88536721236579724695894882620, −2.53960898327844774755503194614, −1.43465512076715108952480991095,
1.43465512076715108952480991095, 2.53960898327844774755503194614, 3.88536721236579724695894882620, 5.26887457404686642434919460938, 6.09819524296858142346616504686, 6.64134267177989658435233209515, 8.138393146796464789814199462339, 9.410920192079793164617988334720, 9.998102159020261865955060941052, 10.82395943167387627638475129219
