| L(s) = 1 | + 2-s + 0.105·3-s + 4-s + 1.78·5-s + 0.105·6-s + 0.955·7-s + 8-s − 2.98·9-s + 1.78·10-s − 2.10·11-s + 0.105·12-s − 3.36·13-s + 0.955·14-s + 0.188·15-s + 16-s − 2.54·17-s − 2.98·18-s − 5.77·19-s + 1.78·20-s + 0.101·21-s − 2.10·22-s + 0.486·23-s + 0.105·24-s − 1.83·25-s − 3.36·26-s − 0.634·27-s + 0.955·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.0611·3-s + 0.5·4-s + 0.796·5-s + 0.0432·6-s + 0.360·7-s + 0.353·8-s − 0.996·9-s + 0.562·10-s − 0.634·11-s + 0.0305·12-s − 0.931·13-s + 0.255·14-s + 0.0487·15-s + 0.250·16-s − 0.616·17-s − 0.704·18-s − 1.32·19-s + 0.398·20-s + 0.0220·21-s − 0.448·22-s + 0.101·23-s + 0.0216·24-s − 0.366·25-s − 0.659·26-s − 0.122·27-s + 0.180·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 0.105T + 3T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 7 | \( 1 - 0.955T + 7T^{2} \) |
| 11 | \( 1 + 2.10T + 11T^{2} \) |
| 13 | \( 1 + 3.36T + 13T^{2} \) |
| 17 | \( 1 + 2.54T + 17T^{2} \) |
| 19 | \( 1 + 5.77T + 19T^{2} \) |
| 23 | \( 1 - 0.486T + 23T^{2} \) |
| 29 | \( 1 - 1.06T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 + 4.18T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + 6.72T + 47T^{2} \) |
| 53 | \( 1 - 9.60T + 53T^{2} \) |
| 59 | \( 1 - 2.80T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 + 5.72T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.080363477083733569362471313710, −7.23455495374043617691083981222, −6.43355174119061705331790263557, −5.74175117317064938880834000399, −5.13365719765408875018296443321, −4.44385976300513038665533929374, −3.34509210248570838694660922995, −2.38802920587320033976757758894, −1.93594070229561955932512479181, 0,
1.93594070229561955932512479181, 2.38802920587320033976757758894, 3.34509210248570838694660922995, 4.44385976300513038665533929374, 5.13365719765408875018296443321, 5.74175117317064938880834000399, 6.43355174119061705331790263557, 7.23455495374043617691083981222, 8.080363477083733569362471313710
