| L(s) = 1 | + 3.76·2-s + 6.16·4-s + 23.6·7-s − 6.92·8-s − 11·11-s − 7.39·13-s + 88.9·14-s − 75.3·16-s + 8.68·17-s − 69.7·19-s − 41.3·22-s − 10.1·23-s − 27.8·26-s + 145.·28-s + 73.2·29-s − 290.·31-s − 228.·32-s + 32.6·34-s + 105.·37-s − 262.·38-s − 40.5·41-s + 77.3·43-s − 67.7·44-s − 38.2·46-s − 472.·47-s + 215.·49-s − 45.5·52-s + ⋯ |
| L(s) = 1 | + 1.33·2-s + 0.770·4-s + 1.27·7-s − 0.305·8-s − 0.301·11-s − 0.157·13-s + 1.69·14-s − 1.17·16-s + 0.123·17-s − 0.841·19-s − 0.401·22-s − 0.0921·23-s − 0.209·26-s + 0.982·28-s + 0.469·29-s − 1.68·31-s − 1.26·32-s + 0.164·34-s + 0.469·37-s − 1.11·38-s − 0.154·41-s + 0.274·43-s − 0.232·44-s − 0.122·46-s − 1.46·47-s + 0.628·49-s − 0.121·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 3.76T + 8T^{2} \) |
| 7 | \( 1 - 23.6T + 343T^{2} \) |
| 13 | \( 1 + 7.39T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.68T + 4.91e3T^{2} \) |
| 19 | \( 1 + 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 73.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 290.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 77.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 472.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 205.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 330.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 931.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 418.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 506.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 612.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 54.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 781.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 531.T + 9.12e5T^{2} \) |
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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
−8.118317832921700573298589410104, −7.33304568780582597426060088346, −6.40070387776346695537497493813, −5.61497713994008519672382193005, −4.91837487161020108513765943130, −4.37735124843775149016248548822, −3.48787369225175766709524233710, −2.46813537405271466369895016293, −1.58657254811156824820093674793, 0,
1.58657254811156824820093674793, 2.46813537405271466369895016293, 3.48787369225175766709524233710, 4.37735124843775149016248548822, 4.91837487161020108513765943130, 5.61497713994008519672382193005, 6.40070387776346695537497493813, 7.33304568780582597426060088346, 8.118317832921700573298589410104
