| L(s) = 1 | − 2·2-s + 2·4-s + 2·7-s − 11-s − 4·13-s − 4·14-s − 4·16-s − 2·17-s + 2·22-s − 23-s + 8·26-s + 4·28-s + 7·31-s + 8·32-s + 4·34-s − 3·37-s + 8·41-s + 6·43-s − 2·44-s + 2·46-s + 8·47-s − 3·49-s − 8·52-s − 6·53-s − 5·59-s + 12·61-s − 14·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.755·7-s − 0.301·11-s − 1.10·13-s − 1.06·14-s − 16-s − 0.485·17-s + 0.426·22-s − 0.208·23-s + 1.56·26-s + 0.755·28-s + 1.25·31-s + 1.41·32-s + 0.685·34-s − 0.493·37-s + 1.24·41-s + 0.914·43-s − 0.301·44-s + 0.294·46-s + 1.16·47-s − 3/7·49-s − 1.10·52-s − 0.824·53-s − 0.650·59-s + 1.53·61-s − 1.77·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7533296616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7533296616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.845671750271850228494101894633, −8.276530745833509741483457469247, −7.56082953278459650643638402038, −7.06748750449497583777914546446, −5.99803243316135624496921094343, −4.90502322677283757606823458951, −4.29108769550776490491760393895, −2.69851752539611099704924785810, −1.91502190096627856445198269376, −0.68132836611621051670216891055,
0.68132836611621051670216891055, 1.91502190096627856445198269376, 2.69851752539611099704924785810, 4.29108769550776490491760393895, 4.90502322677283757606823458951, 5.99803243316135624496921094343, 7.06748750449497583777914546446, 7.56082953278459650643638402038, 8.276530745833509741483457469247, 8.845671750271850228494101894633
