| L(s) = 1 | + 0.755·2-s − 3-s − 1.42·4-s + 3.73·5-s − 0.755·6-s − 2.58·8-s + 9-s + 2.82·10-s + 3.02·11-s + 1.42·12-s − 13-s − 3.73·15-s + 0.903·16-s + 4.32·17-s + 0.755·18-s − 0.0963·19-s − 5.34·20-s + 2.28·22-s − 5.25·23-s + 2.58·24-s + 8.97·25-s − 0.755·26-s − 27-s + 2.88·29-s − 2.82·30-s − 2.71·31-s + 5.86·32-s + ⋯ |
| L(s) = 1 | + 0.533·2-s − 0.577·3-s − 0.714·4-s + 1.67·5-s − 0.308·6-s − 0.915·8-s + 0.333·9-s + 0.892·10-s + 0.911·11-s + 0.412·12-s − 0.277·13-s − 0.965·15-s + 0.225·16-s + 1.04·17-s + 0.177·18-s − 0.0220·19-s − 1.19·20-s + 0.486·22-s − 1.09·23-s + 0.528·24-s + 1.79·25-s − 0.148·26-s − 0.192·27-s + 0.535·29-s − 0.515·30-s − 0.487·31-s + 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.197927254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197927254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.755T + 2T^{2} \) |
| 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 + 0.0963T + 19T^{2} \) |
| 23 | \( 1 + 5.25T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 2.71T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 41 | \( 1 + 9.88T + 41T^{2} \) |
| 43 | \( 1 - 7.57T + 43T^{2} \) |
| 47 | \( 1 - 7.49T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 9.80T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 6.61T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 0.800T + 83T^{2} \) |
| 89 | \( 1 - 1.24T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−9.286194431467633406995001353990, −8.730107799293576551001328247926, −7.49828745601701024172717582682, −6.38679299768546022886121535854, −5.90040738099541726270513430240, −5.31974523404287702614896668634, −4.45550891642839627997309916182, −3.47532105435438818330935910024, −2.20288203940855605129089857032, −1.01260543720156827210262163289,
1.01260543720156827210262163289, 2.20288203940855605129089857032, 3.47532105435438818330935910024, 4.45550891642839627997309916182, 5.31974523404287702614896668634, 5.90040738099541726270513430240, 6.38679299768546022886121535854, 7.49828745601701024172717582682, 8.730107799293576551001328247926, 9.286194431467633406995001353990
