| L(s) = 1 | + 0.277·2-s − 0.494·3-s − 1.92·4-s − 1.23·5-s − 0.137·6-s + 1.36·7-s − 1.08·8-s − 2.75·9-s − 0.342·10-s − 4.69·11-s + 0.951·12-s − 0.0431·13-s + 0.379·14-s + 0.610·15-s + 3.54·16-s − 7.31·17-s − 0.764·18-s − 0.697·19-s + 2.37·20-s − 0.676·21-s − 1.30·22-s + 1.41·23-s + 0.538·24-s − 3.47·25-s − 0.0119·26-s + 2.84·27-s − 2.62·28-s + ⋯ |
| L(s) = 1 | + 0.196·2-s − 0.285·3-s − 0.961·4-s − 0.551·5-s − 0.0560·6-s + 0.516·7-s − 0.384·8-s − 0.918·9-s − 0.108·10-s − 1.41·11-s + 0.274·12-s − 0.0119·13-s + 0.101·14-s + 0.157·15-s + 0.885·16-s − 1.77·17-s − 0.180·18-s − 0.160·19-s + 0.530·20-s − 0.147·21-s − 0.278·22-s + 0.294·23-s + 0.109·24-s − 0.695·25-s − 0.00235·26-s + 0.548·27-s − 0.496·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.277T + 2T^{2} \) |
| 3 | \( 1 + 0.494T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 + 0.0431T + 13T^{2} \) |
| 17 | \( 1 + 7.31T + 17T^{2} \) |
| 19 | \( 1 + 0.697T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.30T + 29T^{2} \) |
| 31 | \( 1 - 3.39T + 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 5.45T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 5.24T + 47T^{2} \) |
| 53 | \( 1 + 8.57T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 1.86T + 71T^{2} \) |
| 73 | \( 1 - 6.47T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 2.23T + 97T^{2} \) |
| 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
−11.56137825020424525513235514748, −10.89163482340058271372820418758, −9.714030430573052451950270538104, −8.408849937708722147073447676074, −8.086314886087834652804515808816, −6.40155979517151252568185227180, −5.13631820784823267630632931308, −4.44359604513990188146003081844, −2.81935360930422956030879563752, 0,
2.81935360930422956030879563752, 4.44359604513990188146003081844, 5.13631820784823267630632931308, 6.40155979517151252568185227180, 8.086314886087834652804515808816, 8.408849937708722147073447676074, 9.714030430573052451950270538104, 10.89163482340058271372820418758, 11.56137825020424525513235514748
