| L(s) = 1 | + 2.55·2-s + 2.76·3-s + 4.50·4-s − 0.117·5-s + 7.05·6-s + 0.649·7-s + 6.39·8-s + 4.64·9-s − 0.300·10-s − 3.37·11-s + 12.4·12-s − 4.83·13-s + 1.65·14-s − 0.325·15-s + 7.29·16-s − 2.19·17-s + 11.8·18-s + 2.67·19-s − 0.530·20-s + 1.79·21-s − 8.61·22-s + 6.36·23-s + 17.6·24-s − 4.98·25-s − 12.3·26-s + 4.54·27-s + 2.92·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 1.59·3-s + 2.25·4-s − 0.0526·5-s + 2.87·6-s + 0.245·7-s + 2.26·8-s + 1.54·9-s − 0.0949·10-s − 1.01·11-s + 3.59·12-s − 1.34·13-s + 0.442·14-s − 0.0840·15-s + 1.82·16-s − 0.531·17-s + 2.79·18-s + 0.614·19-s − 0.118·20-s + 0.391·21-s − 1.83·22-s + 1.32·23-s + 3.60·24-s − 0.997·25-s − 2.41·26-s + 0.874·27-s + 0.553·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.205929679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.205929679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 0.117T + 5T^{2} \) |
| 7 | \( 1 - 0.649T + 7T^{2} \) |
| 11 | \( 1 + 3.37T + 11T^{2} \) |
| 13 | \( 1 + 4.83T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 - 2.67T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 + 0.967T + 29T^{2} \) |
| 31 | \( 1 + 1.96T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 - 5.20T + 41T^{2} \) |
| 47 | \( 1 + 4.11T + 47T^{2} \) |
| 53 | \( 1 + 8.39T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.117T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 0.127T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 6.58T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−9.308409306497939994477779118314, −8.125095952811971968533647971693, −7.54200682799669528594732147609, −6.98639726978163466380951642737, −5.75893866927491930200180995055, −4.88878670693090267902469067025, −4.30148230141702317786300520253, −3.21768968060577321991101075093, −2.69666954921231743597225560206, −1.95790203384400304053843776482,
1.95790203384400304053843776482, 2.69666954921231743597225560206, 3.21768968060577321991101075093, 4.30148230141702317786300520253, 4.88878670693090267902469067025, 5.75893866927491930200180995055, 6.98639726978163466380951642737, 7.54200682799669528594732147609, 8.125095952811971968533647971693, 9.308409306497939994477779118314
