| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 5·11-s − 6·13-s + 16-s + 4·17-s + 4·19-s + 2·20-s + 5·22-s − 4·23-s − 25-s − 6·26-s + 7·29-s − 3·31-s + 32-s + 4·34-s + 8·37-s + 4·38-s + 2·40-s + 6·41-s + 8·43-s + 5·44-s − 4·46-s − 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.50·11-s − 1.66·13-s + 1/4·16-s + 0.970·17-s + 0.917·19-s + 0.447·20-s + 1.06·22-s − 0.834·23-s − 1/5·25-s − 1.17·26-s + 1.29·29-s − 0.538·31-s + 0.176·32-s + 0.685·34-s + 1.31·37-s + 0.648·38-s + 0.316·40-s + 0.937·41-s + 1.21·43-s + 0.753·44-s − 0.589·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.702424393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.702424393\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−9.095167930688383706202169260535, −7.83113443648649543278880518873, −7.29004140217475303196893770682, −6.30075190335325105543587997729, −5.84641046436081104626477867199, −4.92798039462414312972984208057, −4.18862403244071980910959065454, −3.13451126524105340409024501387, −2.24207457127108867855044837032, −1.19305358333700318439184302534,
1.19305358333700318439184302534, 2.24207457127108867855044837032, 3.13451126524105340409024501387, 4.18862403244071980910959065454, 4.92798039462414312972984208057, 5.84641046436081104626477867199, 6.30075190335325105543587997729, 7.29004140217475303196893770682, 7.83113443648649543278880518873, 9.095167930688383706202169260535
