| L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s + 3·11-s − 13-s − 2·14-s + 16-s + 3·17-s + 8·19-s − 20-s − 3·22-s − 3·23-s + 25-s + 26-s + 2·28-s + 9·29-s − 7·31-s − 32-s − 3·34-s − 2·35-s + 2·37-s − 8·38-s + 40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 1.83·19-s − 0.223·20-s − 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.67·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s − 0.338·35-s + 0.328·37-s − 1.29·38-s + 0.158·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9871440961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9871440961\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−11.78318272116545337419358605651, −11.05191214604594379403959723854, −9.898970025925554421885060612926, −9.095398543889443614795538732083, −7.955240080686076886252892838851, −7.36179448350786937184598515633, −6.02650666217825993704928776127, −4.68795794788034775526265746527, −3.21325548378725257481862846236, −1.33830057852400417582257636134,
1.33830057852400417582257636134, 3.21325548378725257481862846236, 4.68795794788034775526265746527, 6.02650666217825993704928776127, 7.36179448350786937184598515633, 7.955240080686076886252892838851, 9.095398543889443614795538732083, 9.898970025925554421885060612926, 11.05191214604594379403959723854, 11.78318272116545337419358605651
