| L(s) = 1 | + 0.326·2-s + 1.71·3-s − 1.89·4-s + 0.560·6-s − 3.42·7-s − 1.27·8-s − 0.0574·9-s − 5.34·11-s − 3.24·12-s + 3.52·13-s − 1.11·14-s + 3.37·16-s − 2.55·17-s − 0.0187·18-s − 2.02·19-s − 5.87·21-s − 1.74·22-s − 7.57·23-s − 2.18·24-s + 1.15·26-s − 5.24·27-s + 6.48·28-s + 4.74·29-s + 1.62·31-s + 3.64·32-s − 9.16·33-s − 0.835·34-s + ⋯ |
| L(s) = 1 | + 0.231·2-s + 0.990·3-s − 0.946·4-s + 0.228·6-s − 1.29·7-s − 0.449·8-s − 0.0191·9-s − 1.61·11-s − 0.937·12-s + 0.976·13-s − 0.299·14-s + 0.842·16-s − 0.620·17-s − 0.00442·18-s − 0.464·19-s − 1.28·21-s − 0.372·22-s − 1.57·23-s − 0.445·24-s + 0.225·26-s − 1.00·27-s + 1.22·28-s + 0.880·29-s + 0.291·31-s + 0.644·32-s − 1.59·33-s − 0.143·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 0.326T + 2T^{2} \) |
| 3 | \( 1 - 1.71T + 3T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 0.0134T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 + 6.94T + 47T^{2} \) |
| 53 | \( 1 + 1.72T + 53T^{2} \) |
| 59 | \( 1 + 0.0221T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 - 0.426T + 79T^{2} \) |
| 83 | \( 1 - 6.04T + 83T^{2} \) |
| 89 | \( 1 + 6.09T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−9.959948851543899065931855056881, −9.279444252474207196521086995259, −8.370960037457355089489503817475, −7.952339574276146121301535559482, −6.42011899175478648977352589721, −5.62016418082582084107205083364, −4.30304068623816507143300134061, −3.37909486244557874191062119698, −2.52815131647055018545135344659, 0,
2.52815131647055018545135344659, 3.37909486244557874191062119698, 4.30304068623816507143300134061, 5.62016418082582084107205083364, 6.42011899175478648977352589721, 7.952339574276146121301535559482, 8.370960037457355089489503817475, 9.279444252474207196521086995259, 9.959948851543899065931855056881
