| L(s) = 1 | − 2-s + 4-s + 3·5-s − 7-s − 8-s − 3·10-s + 11-s − 13-s + 14-s + 16-s + 4·17-s − 4·19-s + 3·20-s − 22-s + 2·23-s + 4·25-s + 26-s − 28-s − 29-s − 31-s − 32-s − 4·34-s − 3·35-s + 6·37-s + 4·38-s − 3·40-s + 3·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.301·11-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.917·19-s + 0.670·20-s − 0.213·22-s + 0.417·23-s + 4/5·25-s + 0.196·26-s − 0.188·28-s − 0.185·29-s − 0.179·31-s − 0.176·32-s − 0.685·34-s − 0.507·35-s + 0.986·37-s + 0.648·38-s − 0.474·40-s + 0.457·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3654 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734267735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734267735\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.788614881121286484317717482185, −7.80737096179843261691350700324, −7.09356113807565322882705387952, −6.22924532291416607536843349075, −5.84685232009221103447646080809, −4.91686369313464712130867824933, −3.74074141322360227356554796872, −2.67506574982067738293442287093, −1.95640360242457362030996671895, −0.872027052872828128451806063497,
0.872027052872828128451806063497, 1.95640360242457362030996671895, 2.67506574982067738293442287093, 3.74074141322360227356554796872, 4.91686369313464712130867824933, 5.84685232009221103447646080809, 6.22924532291416607536843349075, 7.09356113807565322882705387952, 7.80737096179843261691350700324, 8.788614881121286484317717482185
