| L(s) = 1 | + 0.266·2-s − 3-s − 1.92·4-s − 1.23·5-s − 0.266·6-s − 7-s − 1.04·8-s + 9-s − 0.328·10-s + 3.34·11-s + 1.92·12-s + 3.20·13-s − 0.266·14-s + 1.23·15-s + 3.57·16-s − 2.13·17-s + 0.266·18-s − 5.47·19-s + 2.37·20-s + 21-s + 0.891·22-s + 2.53·23-s + 1.04·24-s − 3.47·25-s + 0.853·26-s − 27-s + 1.92·28-s + ⋯ |
| L(s) = 1 | + 0.188·2-s − 0.577·3-s − 0.964·4-s − 0.551·5-s − 0.108·6-s − 0.377·7-s − 0.370·8-s + 0.333·9-s − 0.103·10-s + 1.00·11-s + 0.556·12-s + 0.887·13-s − 0.0712·14-s + 0.318·15-s + 0.894·16-s − 0.517·17-s + 0.0628·18-s − 1.25·19-s + 0.531·20-s + 0.218·21-s + 0.189·22-s + 0.528·23-s + 0.213·24-s − 0.695·25-s + 0.167·26-s − 0.192·27-s + 0.364·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.266T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 + 2.13T + 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 23 | \( 1 - 2.53T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 + 9.34T + 31T^{2} \) |
| 37 | \( 1 - 6.75T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 7.84T + 53T^{2} \) |
| 59 | \( 1 - 3.70T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 + 1.26T + 71T^{2} \) |
| 73 | \( 1 - 4.83T + 73T^{2} \) |
| 79 | \( 1 - 0.908T + 79T^{2} \) |
| 83 | \( 1 + 9.55T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 7.67T + 97T^{2} \) |
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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
−7.35127654504195056771553211049, −6.79062279512486239321787427483, −5.82535328456185671071651935278, −5.63421559976996052119886220690, −4.39883992483863074552147603403, −3.98945286337828885969404652340, −3.57601565344655080337319266753, −2.15977443373201792695305944341, −0.973411414410368980207672024115, 0,
0.973411414410368980207672024115, 2.15977443373201792695305944341, 3.57601565344655080337319266753, 3.98945286337828885969404652340, 4.39883992483863074552147603403, 5.63421559976996052119886220690, 5.82535328456185671071651935278, 6.79062279512486239321787427483, 7.35127654504195056771553211049
