| L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s − 1.23·5-s + 1.61·6-s + 7-s + 2.23·8-s − 2·9-s + 2.00·10-s + 4.47·11-s − 0.618·12-s − 4.23·13-s − 1.61·14-s + 1.23·15-s − 4.85·16-s + 3.23·18-s + 2.76·19-s − 0.763·20-s − 21-s − 7.23·22-s − 2.23·24-s − 3.47·25-s + 6.85·26-s + 5·27-s + 0.618·28-s + 7.47·29-s − 2.00·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.552·5-s + 0.660·6-s + 0.377·7-s + 0.790·8-s − 0.666·9-s + 0.632·10-s + 1.34·11-s − 0.178·12-s − 1.17·13-s − 0.432·14-s + 0.319·15-s − 1.21·16-s + 0.762·18-s + 0.634·19-s − 0.170·20-s − 0.218·21-s − 1.54·22-s − 0.456·24-s − 0.694·25-s + 1.34·26-s + 0.962·27-s + 0.116·28-s + 1.38·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3703 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.76T + 19T^{2} \) |
| 29 | \( 1 - 7.47T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + 7.70T + 37T^{2} \) |
| 41 | \( 1 - 2.23T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 5.47T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 5.76T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 2.47T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.207808107383238833542867151346, −7.50405103179675737014282463025, −6.98526182351502633217908166299, −6.02923659369423103532097790260, −5.09777887713828208758721304428, −4.42720184841194637123442154163, −3.46731387296085484966814456211, −2.15285034112145427455674155984, −1.04454176598662009615372284824, 0,
1.04454176598662009615372284824, 2.15285034112145427455674155984, 3.46731387296085484966814456211, 4.42720184841194637123442154163, 5.09777887713828208758721304428, 6.02923659369423103532097790260, 6.98526182351502633217908166299, 7.50405103179675737014282463025, 8.207808107383238833542867151346
