L(s) = 1 | + 2.52·2-s − 1.80·3-s + 4.39·4-s − 4.56·6-s + 7-s + 6.05·8-s + 0.258·9-s + 3.95·11-s − 7.93·12-s − 0.378·13-s + 2.52·14-s + 6.53·16-s + 3.06·17-s + 0.654·18-s + 3.83·19-s − 1.80·21-s + 10.0·22-s − 23-s − 10.9·24-s − 0.958·26-s + 4.94·27-s + 4.39·28-s − 0.0915·29-s − 8.53·31-s + 4.39·32-s − 7.14·33-s + 7.75·34-s + ⋯ |
L(s) = 1 | + 1.78·2-s − 1.04·3-s + 2.19·4-s − 1.86·6-s + 0.377·7-s + 2.14·8-s + 0.0862·9-s + 1.19·11-s − 2.29·12-s − 0.105·13-s + 0.675·14-s + 1.63·16-s + 0.743·17-s + 0.154·18-s + 0.879·19-s − 0.393·21-s + 2.13·22-s − 0.208·23-s − 2.23·24-s − 0.187·26-s + 0.952·27-s + 0.830·28-s − 0.0169·29-s − 1.53·31-s + 0.777·32-s − 1.24·33-s + 1.32·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.740006969\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.740006969\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + 0.378T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 29 | \( 1 + 0.0915T + 29T^{2} \) |
| 31 | \( 1 + 8.53T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 4.07T + 43T^{2} \) |
| 47 | \( 1 + 4.32T + 47T^{2} \) |
| 53 | \( 1 - 9.57T + 53T^{2} \) |
| 59 | \( 1 - 5.74T + 59T^{2} \) |
| 61 | \( 1 - 2.08T + 61T^{2} \) |
| 67 | \( 1 - 1.88T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 2.71T + 79T^{2} \) |
| 83 | \( 1 + 1.49T + 83T^{2} \) |
| 89 | \( 1 - 2.16T + 89T^{2} \) |
| 97 | \( 1 - 8.04T + 97T^{2} \) |
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
−8.177329760844006325723743243081, −7.22746408680892065234224408842, −6.67028267658882527532655110507, −5.94097921983145758952559813125, −5.39646960457792813351675351511, −4.87621525500550937513068304449, −3.93467415366724349615944950846, −3.36834853212648224527119983192, −2.19336282765768750153327382169, −1.06679172347757760854599377959,
1.06679172347757760854599377959, 2.19336282765768750153327382169, 3.36834853212648224527119983192, 3.93467415366724349615944950846, 4.87621525500550937513068304449, 5.39646960457792813351675351511, 5.94097921983145758952559813125, 6.67028267658882527532655110507, 7.22746408680892065234224408842, 8.177329760844006325723743243081