| L(s) = 1 | + 2.46·3-s − 1.12·5-s + 7-s + 3.05·9-s − 11-s − 13-s − 2.77·15-s + 6.31·17-s − 1.83·19-s + 2.46·21-s − 8.30·23-s − 3.72·25-s + 0.139·27-s − 7.64·29-s − 5.82·31-s − 2.46·33-s − 1.12·35-s + 4.38·37-s − 2.46·39-s − 1.61·41-s − 9.78·43-s − 3.44·45-s + 3.50·47-s + 49-s + 15.5·51-s + 10.6·53-s + 1.12·55-s + ⋯ |
| L(s) = 1 | + 1.42·3-s − 0.504·5-s + 0.377·7-s + 1.01·9-s − 0.301·11-s − 0.277·13-s − 0.717·15-s + 1.53·17-s − 0.420·19-s + 0.537·21-s − 1.73·23-s − 0.745·25-s + 0.0268·27-s − 1.42·29-s − 1.04·31-s − 0.428·33-s − 0.190·35-s + 0.721·37-s − 0.394·39-s − 0.251·41-s − 1.49·43-s − 0.514·45-s + 0.510·47-s + 0.142·49-s + 2.17·51-s + 1.46·53-s + 0.152·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.46T + 3T^{2} \) |
| 5 | \( 1 + 1.12T + 5T^{2} \) |
| 17 | \( 1 - 6.31T + 17T^{2} \) |
| 19 | \( 1 + 1.83T + 19T^{2} \) |
| 23 | \( 1 + 8.30T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 - 4.38T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 - 3.50T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 8.87T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 7.88T + 97T^{2} \) |
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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
−7.46637902570839904548389531002, −7.39844675710195879697804049141, −5.96998906352761119815099785006, −5.47424624731945426026042569783, −4.34535566046405077398806634428, −3.77139960419112355453098144492, −3.20363754886686682504820077493, −2.22471385760089905366997795149, −1.62300830559309975651488187097, 0,
1.62300830559309975651488187097, 2.22471385760089905366997795149, 3.20363754886686682504820077493, 3.77139960419112355453098144492, 4.34535566046405077398806634428, 5.47424624731945426026042569783, 5.96998906352761119815099785006, 7.39844675710195879697804049141, 7.46637902570839904548389531002
