L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s − 2·9-s − 10-s + 12-s + 3·13-s + 14-s + 15-s + 16-s + 2·18-s + 3·19-s + 20-s − 21-s − 7·23-s − 24-s + 25-s − 3·26-s − 5·27-s − 28-s + 8·29-s − 30-s + 4·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.288·12-s + 0.832·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.471·18-s + 0.688·19-s + 0.223·20-s − 0.218·21-s − 1.45·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.962·27-s − 0.188·28-s + 1.48·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.77415679228123013647234359243, −6.61530460288794171614617212198, −6.32157065334701291814715769044, −5.54994033197644939057880939654, −4.65659981718587765978373170381, −3.47855837720200014132152961958, −3.07772874204107042273890327361, −2.13927185923244048035592685239, −1.31152444936769448351580707290, 0,
1.31152444936769448351580707290, 2.13927185923244048035592685239, 3.07772874204107042273890327361, 3.47855837720200014132152961958, 4.65659981718587765978373170381, 5.54994033197644939057880939654, 6.32157065334701291814715769044, 6.61530460288794171614617212198, 7.77415679228123013647234359243