L(s) = 1 | − 2-s + 1.58·3-s + 4-s − 2.70·5-s − 1.58·6-s + 7-s − 8-s − 0.493·9-s + 2.70·10-s − 3.32·11-s + 1.58·12-s − 0.521·13-s − 14-s − 4.27·15-s + 16-s + 5.72·17-s + 0.493·18-s + 1.35·19-s − 2.70·20-s + 1.58·21-s + 3.32·22-s + 0.952·23-s − 1.58·24-s + 2.30·25-s + 0.521·26-s − 5.53·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.914·3-s + 0.5·4-s − 1.20·5-s − 0.646·6-s + 0.377·7-s − 0.353·8-s − 0.164·9-s + 0.854·10-s − 1.00·11-s + 0.457·12-s − 0.144·13-s − 0.267·14-s − 1.10·15-s + 0.250·16-s + 1.38·17-s + 0.116·18-s + 0.311·19-s − 0.604·20-s + 0.345·21-s + 0.708·22-s + 0.198·23-s − 0.323·24-s + 0.460·25-s + 0.102·26-s − 1.06·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.185808111\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.185808111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 + 3.32T + 11T^{2} \) |
| 13 | \( 1 + 0.521T + 13T^{2} \) |
| 17 | \( 1 - 5.72T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 0.952T + 23T^{2} \) |
| 29 | \( 1 - 1.32T + 29T^{2} \) |
| 31 | \( 1 + 5.57T + 31T^{2} \) |
| 37 | \( 1 + 0.895T + 37T^{2} \) |
| 41 | \( 1 - 8.53T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 9.76T + 53T^{2} \) |
| 59 | \( 1 + 0.914T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 + 4.33T + 67T^{2} \) |
| 71 | \( 1 - 5.95T + 71T^{2} \) |
| 73 | \( 1 + 5.02T + 73T^{2} \) |
| 79 | \( 1 - 0.371T + 79T^{2} \) |
| 83 | \( 1 - 6.44T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 7.32T + 97T^{2} \) |
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
−8.031965311843290808499860545306, −7.59631797617446937067810124981, −7.25035075483357765880208223988, −5.92090146320424831716452300807, −5.26020590843862247321496138981, −4.24330170367264792317037675053, −3.35278633057662975517992249521, −2.89016806522973971402369010887, −1.87231099767934115351034939466, −0.59248506103041994353277359997,
0.59248506103041994353277359997, 1.87231099767934115351034939466, 2.89016806522973971402369010887, 3.35278633057662975517992249521, 4.24330170367264792317037675053, 5.26020590843862247321496138981, 5.92090146320424831716452300807, 7.25035075483357765880208223988, 7.59631797617446937067810124981, 8.031965311843290808499860545306