| L(s) = 1 | + 1.23·2-s − 2.02·3-s − 0.482·4-s + 2.63·5-s − 2.49·6-s + 3.75·7-s − 3.05·8-s + 1.11·9-s + 3.24·10-s − 11-s + 0.978·12-s − 1.64·13-s + 4.62·14-s − 5.33·15-s − 2.80·16-s − 1.41·17-s + 1.36·18-s − 1.26·20-s − 7.62·21-s − 1.23·22-s − 2.42·23-s + 6.20·24-s + 1.92·25-s − 2.03·26-s + 3.82·27-s − 1.81·28-s + 7.21·29-s + ⋯ |
| L(s) = 1 | + 0.871·2-s − 1.17·3-s − 0.241·4-s + 1.17·5-s − 1.01·6-s + 1.42·7-s − 1.08·8-s + 0.370·9-s + 1.02·10-s − 0.301·11-s + 0.282·12-s − 0.457·13-s + 1.23·14-s − 1.37·15-s − 0.700·16-s − 0.342·17-s + 0.322·18-s − 0.283·20-s − 1.66·21-s − 0.262·22-s − 0.505·23-s + 1.26·24-s + 0.384·25-s − 0.398·26-s + 0.736·27-s − 0.342·28-s + 1.33·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.23T + 2T^{2} \) |
| 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 - 3.75T + 7T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 + 9.66T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 - 0.0866T + 47T^{2} \) |
| 53 | \( 1 + 1.19T + 53T^{2} \) |
| 59 | \( 1 + 2.72T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 4.15T + 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 3.17T + 79T^{2} \) |
| 83 | \( 1 + 1.23T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−8.234609058282839573893145176082, −6.95273130391521232708365123988, −6.37745555182308308932353118086, −5.36077469066162409311095072548, −5.30667442258187139649416571842, −4.73556571190055223461658466814, −3.66454799691653072544041252701, −2.40228142487565428710539188251, −1.53159151173408042787073134414, 0,
1.53159151173408042787073134414, 2.40228142487565428710539188251, 3.66454799691653072544041252701, 4.73556571190055223461658466814, 5.30667442258187139649416571842, 5.36077469066162409311095072548, 6.37745555182308308932353118086, 6.95273130391521232708365123988, 8.234609058282839573893145176082
