| L(s) = 1 | − 2.12·2-s + 9.87·3-s − 3.49·4-s − 15.5·5-s − 20.9·6-s − 16.3·7-s + 24.3·8-s + 70.4·9-s + 33.0·10-s + 42.7·11-s − 34.5·12-s − 44.4·13-s + 34.7·14-s − 153.·15-s − 23.8·16-s − 80.6·17-s − 149.·18-s + 9.72·19-s + 54.4·20-s − 161.·21-s − 90.6·22-s + 81.9·23-s + 240.·24-s + 117.·25-s + 94.2·26-s + 428.·27-s + 57.2·28-s + ⋯ |
| L(s) = 1 | − 0.750·2-s + 1.89·3-s − 0.436·4-s − 1.39·5-s − 1.42·6-s − 0.884·7-s + 1.07·8-s + 2.60·9-s + 1.04·10-s + 1.17·11-s − 0.829·12-s − 0.947·13-s + 0.663·14-s − 2.64·15-s − 0.372·16-s − 1.15·17-s − 1.95·18-s + 0.117·19-s + 0.608·20-s − 1.68·21-s − 0.878·22-s + 0.743·23-s + 2.04·24-s + 0.940·25-s + 0.711·26-s + 3.05·27-s + 0.386·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.12T + 8T^{2} \) |
| 3 | \( 1 - 9.87T + 27T^{2} \) |
| 5 | \( 1 + 15.5T + 125T^{2} \) |
| 7 | \( 1 + 16.3T + 343T^{2} \) |
| 11 | \( 1 - 42.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 80.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.72T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 170.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 115.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 56.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 9.51T + 6.89e4T^{2} \) |
| 47 | \( 1 - 9.55T + 1.03e5T^{2} \) |
| 53 | \( 1 - 254.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 288.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 67.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 273.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 113.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 908.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.27e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.39e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 253.T + 9.12e5T^{2} \) |
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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
−8.722671771154836271788983009336, −7.914762801409641537455163272345, −7.20918157355800132080772028592, −6.78455492822714288545038116882, −4.66327602085302054830268991575, −4.11141472374008668978097045314, −3.44574481407593562954260157524, −2.51592920621946336735847834057, −1.21560243758786520033869372159, 0,
1.21560243758786520033869372159, 2.51592920621946336735847834057, 3.44574481407593562954260157524, 4.11141472374008668978097045314, 4.66327602085302054830268991575, 6.78455492822714288545038116882, 7.20918157355800132080772028592, 7.914762801409641537455163272345, 8.722671771154836271788983009336
