| L(s) = 1 | + 0.540·3-s + 5-s − 1.71·7-s − 2.70·9-s − 1.80·11-s − 1.10·13-s + 0.540·15-s + 2.19·17-s + 0.480·19-s − 0.926·21-s + 7.44·23-s + 25-s − 3.08·27-s + 7.46·29-s − 2.62·31-s − 0.977·33-s − 1.71·35-s + 5.11·37-s − 0.598·39-s − 9.19·41-s − 4.50·43-s − 2.70·45-s − 3.42·47-s − 4.06·49-s + 1.18·51-s − 6.58·53-s − 1.80·55-s + ⋯ |
| L(s) = 1 | + 0.312·3-s + 0.447·5-s − 0.647·7-s − 0.902·9-s − 0.545·11-s − 0.306·13-s + 0.139·15-s + 0.532·17-s + 0.110·19-s − 0.202·21-s + 1.55·23-s + 0.200·25-s − 0.593·27-s + 1.38·29-s − 0.471·31-s − 0.170·33-s − 0.289·35-s + 0.841·37-s − 0.0957·39-s − 1.43·41-s − 0.687·43-s − 0.403·45-s − 0.500·47-s − 0.580·49-s + 0.166·51-s − 0.904·53-s − 0.243·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.540T + 3T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 - 0.480T + 19T^{2} \) |
| 23 | \( 1 - 7.44T + 23T^{2} \) |
| 29 | \( 1 - 7.46T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 5.11T + 37T^{2} \) |
| 41 | \( 1 + 9.19T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 + 3.42T + 47T^{2} \) |
| 53 | \( 1 + 6.58T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 4.35T + 61T^{2} \) |
| 67 | \( 1 + 0.770T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−7.54032366772300740467229879613, −6.66006805104819422928468520366, −6.25086974952215809123866487393, −5.17121261860218925498527606263, −5.02400098291232325445380677295, −3.64618757362354741050986592978, −2.98944489230418404864526087863, −2.49749746370703838003626127141, −1.26675013991648077795279158780, 0,
1.26675013991648077795279158780, 2.49749746370703838003626127141, 2.98944489230418404864526087863, 3.64618757362354741050986592978, 5.02400098291232325445380677295, 5.17121261860218925498527606263, 6.25086974952215809123866487393, 6.66006805104819422928468520366, 7.54032366772300740467229879613
