| L(s) = 1 | + 2.23·2-s + 2.39·3-s + 2.97·4-s − 5-s + 5.33·6-s − 7-s + 2.17·8-s + 2.72·9-s − 2.23·10-s + 7.12·12-s + 2.48·13-s − 2.23·14-s − 2.39·15-s − 1.09·16-s + 4.59·17-s + 6.08·18-s + 6.44·19-s − 2.97·20-s − 2.39·21-s + 3.74·23-s + 5.20·24-s + 25-s + 5.53·26-s − 0.653·27-s − 2.97·28-s − 5.59·29-s − 5.33·30-s + ⋯ |
| L(s) = 1 | + 1.57·2-s + 1.38·3-s + 1.48·4-s − 0.447·5-s + 2.17·6-s − 0.377·7-s + 0.769·8-s + 0.908·9-s − 0.705·10-s + 2.05·12-s + 0.688·13-s − 0.596·14-s − 0.617·15-s − 0.274·16-s + 1.11·17-s + 1.43·18-s + 1.47·19-s − 0.665·20-s − 0.522·21-s + 0.780·23-s + 1.06·24-s + 0.200·25-s + 1.08·26-s − 0.125·27-s − 0.562·28-s − 1.03·29-s − 0.974·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.494336759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.494336759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 - 2.39T + 3T^{2} \) |
| 13 | \( 1 - 2.48T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 31 | \( 1 - 7.48T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 - 1.74T + 41T^{2} \) |
| 43 | \( 1 - 4.05T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 - 0.903T + 71T^{2} \) |
| 73 | \( 1 - 0.771T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 - 4.58T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.148347234691078939552276167639, −7.63857039649509054176106971064, −6.94028562829314260500894665141, −5.97773399881703914946097217669, −5.36176718046844257193615006377, −4.37386749815695823173131707372, −3.68024066732389953269494050081, −3.12387750033167710660828385979, −2.65037427453780384250904573448, −1.27373533064874860275732586555,
1.27373533064874860275732586555, 2.65037427453780384250904573448, 3.12387750033167710660828385979, 3.68024066732389953269494050081, 4.37386749815695823173131707372, 5.36176718046844257193615006377, 5.97773399881703914946097217669, 6.94028562829314260500894665141, 7.63857039649509054176106971064, 8.148347234691078939552276167639
