| L(s) = 1 | + 0.146·2-s − 3-s − 1.97·4-s + 1.04·5-s − 0.146·6-s − 7-s − 0.581·8-s + 9-s + 0.152·10-s − 3.53·11-s + 1.97·12-s − 0.146·14-s − 1.04·15-s + 3.87·16-s + 6.85·17-s + 0.146·18-s + 3.55·19-s − 2.06·20-s + 21-s − 0.516·22-s + 0.0937·23-s + 0.581·24-s − 3.91·25-s − 27-s + 1.97·28-s − 10.5·29-s − 0.152·30-s + ⋯ |
| L(s) = 1 | + 0.103·2-s − 0.577·3-s − 0.989·4-s + 0.466·5-s − 0.0596·6-s − 0.377·7-s − 0.205·8-s + 0.333·9-s + 0.0482·10-s − 1.06·11-s + 0.571·12-s − 0.0390·14-s − 0.269·15-s + 0.968·16-s + 1.66·17-s + 0.0344·18-s + 0.814·19-s − 0.461·20-s + 0.218·21-s − 0.110·22-s + 0.0195·23-s + 0.118·24-s − 0.782·25-s − 0.192·27-s + 0.373·28-s − 1.96·29-s − 0.0278·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9572521572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9572521572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.146T + 2T^{2} \) |
| 5 | \( 1 - 1.04T + 5T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 0.0937T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 - 9.55T + 47T^{2} \) |
| 53 | \( 1 + 3.01T + 53T^{2} \) |
| 59 | \( 1 + 0.747T + 59T^{2} \) |
| 61 | \( 1 - 5.72T + 61T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + 4.41T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 0.520T + 83T^{2} \) |
| 89 | \( 1 - 4.78T + 89T^{2} \) |
| 97 | \( 1 + 5.89T + 97T^{2} \) |
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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
−8.661501795411068949268154921949, −7.62922763841549387507879352442, −7.32431303684221376086393565619, −5.87758025674721062139913443194, −5.55056839639382088994214401441, −5.07154200841789528057426898723, −3.83788527711345913435509997033, −3.27804776965463905262742928138, −1.88658718603194916639427147836, −0.58402048596228891002918675683,
0.58402048596228891002918675683, 1.88658718603194916639427147836, 3.27804776965463905262742928138, 3.83788527711345913435509997033, 5.07154200841789528057426898723, 5.55056839639382088994214401441, 5.87758025674721062139913443194, 7.32431303684221376086393565619, 7.62922763841549387507879352442, 8.661501795411068949268154921949
