| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.79·7-s + 8-s + 9-s + 10-s − 0.224·11-s + 12-s − 0.158·13-s + 3.79·14-s + 15-s + 16-s + 18-s + 4.76·19-s + 20-s + 3.79·21-s − 0.224·22-s + 0.585·23-s + 24-s + 25-s − 0.158·26-s + 27-s + 3.79·28-s − 2.22·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.43·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.0677·11-s + 0.288·12-s − 0.0440·13-s + 1.01·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s + 1.09·19-s + 0.223·20-s + 0.828·21-s − 0.0479·22-s + 0.122·23-s + 0.204·24-s + 0.200·25-s − 0.0311·26-s + 0.192·27-s + 0.717·28-s − 0.413·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.080833293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.080833293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 + 0.224T + 11T^{2} \) |
| 13 | \( 1 + 0.158T + 13T^{2} \) |
| 19 | \( 1 - 4.76T + 19T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 8.78T + 31T^{2} \) |
| 37 | \( 1 - 2.76T + 37T^{2} \) |
| 41 | \( 1 - 0.712T + 41T^{2} \) |
| 43 | \( 1 + 3.21T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 + 9.19T + 53T^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 - 0.383T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 + 0.424T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−7.72356546692076808115090629817, −7.19632990543968982132379810590, −6.27500209678043858551241620505, −5.60010641416813494707174625094, −4.77666317328498954749136579914, −4.51186014860809214521652434544, −3.39434925941998593506331027837, −2.72881107533047356500285913311, −1.83892015386933369345123301306, −1.17322488347607105263956957588,
1.17322488347607105263956957588, 1.83892015386933369345123301306, 2.72881107533047356500285913311, 3.39434925941998593506331027837, 4.51186014860809214521652434544, 4.77666317328498954749136579914, 5.60010641416813494707174625094, 6.27500209678043858551241620505, 7.19632990543968982132379810590, 7.72356546692076808115090629817
