| L(s) = 1 | + 2-s − 3-s + 4-s + 0.330·5-s − 6-s − 7-s + 8-s + 9-s + 0.330·10-s + 2.17·11-s − 12-s − 4.45·13-s − 14-s − 0.330·15-s + 16-s + 1.78·17-s + 18-s + 2.91·19-s + 0.330·20-s + 21-s + 2.17·22-s + 3.78·23-s − 24-s − 4.89·25-s − 4.45·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.147·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.104·10-s + 0.654·11-s − 0.288·12-s − 1.23·13-s − 0.267·14-s − 0.0852·15-s + 0.250·16-s + 0.432·17-s + 0.235·18-s + 0.667·19-s + 0.0738·20-s + 0.218·21-s + 0.463·22-s + 0.789·23-s − 0.204·24-s − 0.978·25-s − 0.873·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 0.330T + 5T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 2.91T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + 8.20T + 37T^{2} \) |
| 41 | \( 1 - 1.75T + 41T^{2} \) |
| 43 | \( 1 + 8.68T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 0.729T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 4.37T + 97T^{2} \) |
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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
−7.27779946778901313388211856061, −6.77150708662151599734665403660, −5.97002380066278327392846730988, −5.33965503727688468902840668175, −4.86273001973834886779148638310, −3.86647306356166603968880054970, −3.31627504696051612711732813820, −2.27060669908548120082801698813, −1.38214743603594553681469441532, 0,
1.38214743603594553681469441532, 2.27060669908548120082801698813, 3.31627504696051612711732813820, 3.86647306356166603968880054970, 4.86273001973834886779148638310, 5.33965503727688468902840668175, 5.97002380066278327392846730988, 6.77150708662151599734665403660, 7.27779946778901313388211856061
