| L(s) = 1 | + 1.92·3-s + 5-s + 3.41·7-s + 0.717·9-s + 4.88·11-s − 1.11·13-s + 1.92·15-s − 3.64·17-s + 1.58·19-s + 6.58·21-s − 8.13·23-s + 25-s − 4.40·27-s + 29-s + 6.71·31-s + 9.42·33-s + 3.41·35-s + 2.06·37-s − 2.15·39-s + 5.19·41-s + 10.1·43-s + 0.717·45-s + 8.48·47-s + 4.68·49-s − 7.01·51-s − 10.8·53-s + 4.88·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 0.447·5-s + 1.29·7-s + 0.239·9-s + 1.47·11-s − 0.309·13-s + 0.497·15-s − 0.882·17-s + 0.364·19-s + 1.43·21-s − 1.69·23-s + 0.200·25-s − 0.846·27-s + 0.185·29-s + 1.20·31-s + 1.64·33-s + 0.577·35-s + 0.339·37-s − 0.344·39-s + 0.810·41-s + 1.55·43-s + 0.106·45-s + 1.23·47-s + 0.668·49-s − 0.982·51-s − 1.49·53-s + 0.659·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.465567293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.465567293\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.92T + 3T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 1.11T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 - 1.58T + 19T^{2} \) |
| 23 | \( 1 + 8.13T + 23T^{2} \) |
| 31 | \( 1 - 6.71T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 - 5.19T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 7.50T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + 4.83T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.79532318879636238237173404169, −7.27294972054433684934795081504, −6.24484651471766296152635008020, −5.83986401640519521977036952736, −4.62026372314784534183114862744, −4.29789819656543074556240088579, −3.43857626760359377026770431658, −2.37659138727350784648141047916, −1.98346944974395038378352768445, −1.01141507522903470635733248117,
1.01141507522903470635733248117, 1.98346944974395038378352768445, 2.37659138727350784648141047916, 3.43857626760359377026770431658, 4.29789819656543074556240088579, 4.62026372314784534183114862744, 5.83986401640519521977036952736, 6.24484651471766296152635008020, 7.27294972054433684934795081504, 7.79532318879636238237173404169
