| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s − 5·7-s + 8-s + 9-s + 4·10-s + 11-s − 12-s + 13-s − 5·14-s − 4·15-s + 16-s + 2·17-s + 18-s + 4·20-s + 5·21-s + 22-s + 23-s − 24-s + 11·25-s + 26-s − 27-s − 5·28-s − 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 1.33·14-s − 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.894·20-s + 1.09·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s − 0.192·27-s − 0.944·28-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.308817380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.308817380\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.43171675687896454842470843990, −6.68884411370091290550212615228, −6.35685957923522148700389960305, −5.75980818202097717509072189208, −5.37997953617997155666508386074, −4.34220612666453196994806930554, −3.40984233196082597771477573403, −2.79070553912495690705951621981, −1.90187334590119711620845546363, −0.836808899007564406037717788909,
0.836808899007564406037717788909, 1.90187334590119711620845546363, 2.79070553912495690705951621981, 3.40984233196082597771477573403, 4.34220612666453196994806930554, 5.37997953617997155666508386074, 5.75980818202097717509072189208, 6.35685957923522148700389960305, 6.68884411370091290550212615228, 7.43171675687896454842470843990
