| L(s) = 1 | − 1.78·2-s + 2.96·3-s + 1.18·4-s − 0.596·5-s − 5.29·6-s − 4.36·7-s + 1.45·8-s + 5.80·9-s + 1.06·10-s + 2.03·11-s + 3.51·12-s − 1.74·13-s + 7.78·14-s − 1.76·15-s − 4.96·16-s + 1.46·17-s − 10.3·18-s − 2.17·19-s − 0.705·20-s − 12.9·21-s − 3.63·22-s − 2.72·23-s + 4.32·24-s − 4.64·25-s + 3.11·26-s + 8.31·27-s − 5.16·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 1.71·3-s + 0.591·4-s − 0.266·5-s − 2.16·6-s − 1.64·7-s + 0.515·8-s + 1.93·9-s + 0.336·10-s + 0.613·11-s + 1.01·12-s − 0.483·13-s + 2.08·14-s − 0.456·15-s − 1.24·16-s + 0.355·17-s − 2.43·18-s − 0.499·19-s − 0.157·20-s − 2.82·21-s − 0.774·22-s − 0.567·23-s + 0.882·24-s − 0.928·25-s + 0.609·26-s + 1.59·27-s − 0.975·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.78T + 2T^{2} \) |
| 3 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 + 0.596T + 5T^{2} \) |
| 7 | \( 1 + 4.36T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 7.57T + 29T^{2} \) |
| 31 | \( 1 + 0.569T + 31T^{2} \) |
| 37 | \( 1 + 3.96T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 47 | \( 1 - 9.36T + 47T^{2} \) |
| 53 | \( 1 + 5.90T + 53T^{2} \) |
| 59 | \( 1 - 5.11T + 59T^{2} \) |
| 61 | \( 1 + 9.78T + 61T^{2} \) |
| 67 | \( 1 + 0.530T + 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 6.03T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.076849809762783086120263587680, −8.249469313041513063642815781337, −7.47821534180616553234907173479, −7.05323794102677076551326680130, −5.96616421252564531868349675907, −4.21526648307800414180937853644, −3.63837075806775440711775301001, −2.66126211719588865621003286539, −1.68194573184341347753522188222, 0,
1.68194573184341347753522188222, 2.66126211719588865621003286539, 3.63837075806775440711775301001, 4.21526648307800414180937853644, 5.96616421252564531868349675907, 7.05323794102677076551326680130, 7.47821534180616553234907173479, 8.249469313041513063642815781337, 9.076849809762783086120263587680
