L(s) = 1 | − 2.29·2-s + 0.376·3-s + 3.28·4-s + 5-s − 0.864·6-s − 1.95·7-s − 2.94·8-s − 2.85·9-s − 2.29·10-s + 2.60·11-s + 1.23·12-s − 2.36·13-s + 4.48·14-s + 0.376·15-s + 0.213·16-s − 1.73·17-s + 6.57·18-s + 1.91·19-s + 3.28·20-s − 0.733·21-s − 5.98·22-s − 4.02·23-s − 1.10·24-s + 25-s + 5.43·26-s − 2.20·27-s − 6.40·28-s + ⋯ |
L(s) = 1 | − 1.62·2-s + 0.217·3-s + 1.64·4-s + 0.447·5-s − 0.352·6-s − 0.737·7-s − 1.04·8-s − 0.952·9-s − 0.726·10-s + 0.785·11-s + 0.356·12-s − 0.656·13-s + 1.19·14-s + 0.0971·15-s + 0.0533·16-s − 0.421·17-s + 1.54·18-s + 0.438·19-s + 0.734·20-s − 0.160·21-s − 1.27·22-s − 0.838·23-s − 0.226·24-s + 0.200·25-s + 1.06·26-s − 0.424·27-s − 1.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5462545848\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5462545848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 3 | \( 1 - 0.376T + 3T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 17 | \( 1 + 1.73T + 17T^{2} \) |
| 19 | \( 1 - 1.91T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 4.76T + 31T^{2} \) |
| 37 | \( 1 + 9.22T + 37T^{2} \) |
| 41 | \( 1 + 5.94T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 0.872T + 47T^{2} \) |
| 53 | \( 1 + 2.14T + 53T^{2} \) |
| 59 | \( 1 - 9.38T + 59T^{2} \) |
| 61 | \( 1 - 0.990T + 61T^{2} \) |
| 67 | \( 1 + 3.56T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 0.651T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 - 2.88T + 97T^{2} \) |
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
−8.066822048228431845428649613152, −7.21407543586856846722742099476, −6.70529262034841345563063115438, −6.05531715534423985856095383272, −5.27750196289726824920429010048, −4.13487010881505186237865195627, −3.14438680606067030862392128458, −2.41035737117427312577798977261, −1.63657321442414633740673940228, −0.45491894571667527450908018974,
0.45491894571667527450908018974, 1.63657321442414633740673940228, 2.41035737117427312577798977261, 3.14438680606067030862392128458, 4.13487010881505186237865195627, 5.27750196289726824920429010048, 6.05531715534423985856095383272, 6.70529262034841345563063115438, 7.21407543586856846722742099476, 8.066822048228431845428649613152