| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 11-s − 12-s − 13-s + 16-s − 17-s − 18-s − 4·19-s − 22-s + 24-s + 26-s − 27-s + 6·29-s − 8·31-s − 32-s − 33-s + 34-s + 36-s + 2·37-s + 4·38-s + 39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.213·22-s + 0.204·24-s + 0.196·26-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.174·33-s + 0.171·34-s + 1/6·36-s + 0.328·37-s + 0.648·38-s + 0.160·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4415427669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4415427669\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.50273397175418, −11.83010862048664, −11.65857628170483, −11.08124409788259, −10.71912774898243, −10.18004896994949, −9.943809636259665, −9.326952499940793, −8.868054063553232, −8.469902471558687, −7.988216359860873, −7.443637856923625, −6.860961447502799, −6.690623070973913, −6.008901196272999, −5.727906258284097, −4.914758129151618, −4.625190402756134, −4.029405294976964, −3.289460965086682, −2.929756187744726, −1.936003245784190, −1.814495248144630, −0.9943778727441186, −0.2225313467462664,
0.2225313467462664, 0.9943778727441186, 1.814495248144630, 1.936003245784190, 2.929756187744726, 3.289460965086682, 4.029405294976964, 4.625190402756134, 4.914758129151618, 5.727906258284097, 6.008901196272999, 6.690623070973913, 6.860961447502799, 7.443637856923625, 7.988216359860873, 8.469902471558687, 8.868054063553232, 9.326952499940793, 9.943809636259665, 10.18004896994949, 10.71912774898243, 11.08124409788259, 11.65857628170483, 11.83010862048664, 12.50273397175418
