| L(s) = 1 | − 1.62·2-s − 2.62·3-s + 0.624·4-s − 2.73·5-s + 4.25·6-s + 0.785·7-s + 2.22·8-s + 3.91·9-s + 4.43·10-s + 11-s − 1.64·12-s + 2.10·13-s − 1.27·14-s + 7.19·15-s − 4.85·16-s + 1.06·17-s − 6.33·18-s + 6.41·19-s − 1.70·20-s − 2.06·21-s − 1.62·22-s + 7.44·23-s − 5.85·24-s + 2.48·25-s − 3.41·26-s − 2.39·27-s + 0.490·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 1.51·3-s + 0.312·4-s − 1.22·5-s + 1.73·6-s + 0.296·7-s + 0.787·8-s + 1.30·9-s + 1.40·10-s + 0.301·11-s − 0.473·12-s + 0.584·13-s − 0.340·14-s + 1.85·15-s − 1.21·16-s + 0.257·17-s − 1.49·18-s + 1.47·19-s − 0.382·20-s − 0.450·21-s − 0.345·22-s + 1.55·23-s − 1.19·24-s + 0.497·25-s − 0.669·26-s − 0.460·27-s + 0.0927·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 0.785T + 7T^{2} \) |
| 13 | \( 1 - 2.10T + 13T^{2} \) |
| 17 | \( 1 - 1.06T + 17T^{2} \) |
| 19 | \( 1 - 6.41T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 9.99T + 41T^{2} \) |
| 43 | \( 1 + 9.22T + 43T^{2} \) |
| 47 | \( 1 - 0.760T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 7.43T + 59T^{2} \) |
| 61 | \( 1 - 2.54T + 61T^{2} \) |
| 67 | \( 1 + 8.12T + 67T^{2} \) |
| 71 | \( 1 - 5.44T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.52T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 - 2.55T + 89T^{2} \) |
| 97 | \( 1 + 1.11T + 97T^{2} \) |
| 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
−7.49751948929266410562634259601, −6.89636288252607737419718887079, −6.14375401707165555527416158892, −5.18970260556373274952340502109, −4.80215704362956738442347799473, −3.96379401610689562805936809197, −3.13827132014791793825572715279, −1.45589079543160588443177337803, −0.909915253141270340855877518246, 0,
0.909915253141270340855877518246, 1.45589079543160588443177337803, 3.13827132014791793825572715279, 3.96379401610689562805936809197, 4.80215704362956738442347799473, 5.18970260556373274952340502109, 6.14375401707165555527416158892, 6.89636288252607737419718887079, 7.49751948929266410562634259601
