| L(s) = 1 | − 0.342·2-s + 2.18·3-s − 1.88·4-s − 0.548·5-s − 0.747·6-s + 1.82·7-s + 1.32·8-s + 1.77·9-s + 0.187·10-s + 5.99·11-s − 4.11·12-s + 3.70·13-s − 0.624·14-s − 1.19·15-s + 3.31·16-s − 1.64·17-s − 0.607·18-s − 3.15·19-s + 1.03·20-s + 3.98·21-s − 2.05·22-s − 5.46·23-s + 2.90·24-s − 4.69·25-s − 1.26·26-s − 2.67·27-s − 3.43·28-s + ⋯ |
| L(s) = 1 | − 0.241·2-s + 1.26·3-s − 0.941·4-s − 0.245·5-s − 0.305·6-s + 0.689·7-s + 0.469·8-s + 0.591·9-s + 0.0593·10-s + 1.80·11-s − 1.18·12-s + 1.02·13-s − 0.166·14-s − 0.309·15-s + 0.827·16-s − 0.398·17-s − 0.143·18-s − 0.724·19-s + 0.231·20-s + 0.870·21-s − 0.437·22-s − 1.13·23-s + 0.592·24-s − 0.939·25-s − 0.248·26-s − 0.515·27-s − 0.649·28-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
| \(L(1)\) |
\(\approx\) |
\(1.441706713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.441706713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.342T + 2T^{2} \) |
| 3 | \( 1 - 2.18T + 3T^{2} \) |
| 5 | \( 1 + 0.548T + 5T^{2} \) |
| 7 | \( 1 - 1.82T + 7T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 19 | \( 1 + 3.15T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 7.24T + 29T^{2} \) |
| 31 | \( 1 + 9.41T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 + 5.81T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 1.03T + 61T^{2} \) |
| 67 | \( 1 + 8.39T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 6.25T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 + 2.22T + 97T^{2} \) |
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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
−12.17097568843909448698077350577, −11.13436150852445937529285584306, −9.860159602397600924229781343780, −8.809640576770029055604653787701, −8.619301371124561283042832881173, −7.56679588177021050931961990735, −6.06139913893043301591385313644, −4.27020339889448310620982002111, −3.71495710948685740723226854163, −1.68575028161273777703363241585,
1.68575028161273777703363241585, 3.71495710948685740723226854163, 4.27020339889448310620982002111, 6.06139913893043301591385313644, 7.56679588177021050931961990735, 8.619301371124561283042832881173, 8.809640576770029055604653787701, 9.860159602397600924229781343780, 11.13436150852445937529285584306, 12.17097568843909448698077350577
