| L(s) = 1 | − 2-s + 3-s + 4-s + 1.27·5-s − 6-s + 3.55·7-s − 8-s + 9-s − 1.27·10-s − 5.15·11-s + 12-s − 7.00·13-s − 3.55·14-s + 1.27·15-s + 16-s + 1.35·17-s − 18-s − 2.38·19-s + 1.27·20-s + 3.55·21-s + 5.15·22-s − 23-s − 24-s − 3.38·25-s + 7.00·26-s + 27-s + 3.55·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.568·5-s − 0.408·6-s + 1.34·7-s − 0.353·8-s + 0.333·9-s − 0.401·10-s − 1.55·11-s + 0.288·12-s − 1.94·13-s − 0.950·14-s + 0.328·15-s + 0.250·16-s + 0.327·17-s − 0.235·18-s − 0.547·19-s + 0.284·20-s + 0.776·21-s + 1.09·22-s − 0.208·23-s − 0.204·24-s − 0.676·25-s + 1.37·26-s + 0.192·27-s + 0.672·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.27T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 + 7.00T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 2.38T + 19T^{2} \) |
| 31 | \( 1 - 2.05T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 - 2.66T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.79T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 - 4.67T + 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 + 8.29T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 3.38T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
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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
−8.177716014502622897596070605868, −7.53349224104440351118371183534, −7.01718048901894161665593144590, −5.75282622957910978862485857535, −5.08397759792384556307013841825, −4.47017289927552228781068451441, −2.97668329806082001461384336999, −2.28069856641972086804954270467, −1.67739205627273831829708212881, 0,
1.67739205627273831829708212881, 2.28069856641972086804954270467, 2.97668329806082001461384336999, 4.47017289927552228781068451441, 5.08397759792384556307013841825, 5.75282622957910978862485857535, 7.01718048901894161665593144590, 7.53349224104440351118371183534, 8.177716014502622897596070605868
