L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 2·11-s + 3·13-s − 14-s + 16-s − 17-s + 2·19-s + 20-s + 2·22-s + 2·23-s − 4·25-s + 3·26-s − 28-s + 7·29-s + 6·31-s + 32-s − 34-s − 35-s − 7·37-s + 2·38-s + 40-s + 6·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s + 0.223·20-s + 0.426·22-s + 0.417·23-s − 4/5·25-s + 0.588·26-s − 0.188·28-s + 1.29·29-s + 1.07·31-s + 0.176·32-s − 0.171·34-s − 0.169·35-s − 1.15·37-s + 0.324·38-s + 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.846174599\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.846174599\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.909828521481332510569176360142, −9.030249616137121850800328213324, −8.187493256811103457585421215303, −7.03415492686924317997509377431, −6.35204749634029384423645042936, −5.64835753574115072695741973282, −4.57877957875718122510327074450, −3.65183042729088600294599514392, −2.66134844455756994322024129146, −1.30688409677236621634556725692,
1.30688409677236621634556725692, 2.66134844455756994322024129146, 3.65183042729088600294599514392, 4.57877957875718122510327074450, 5.64835753574115072695741973282, 6.35204749634029384423645042936, 7.03415492686924317997509377431, 8.187493256811103457585421215303, 9.030249616137121850800328213324, 9.909828521481332510569176360142