| L(s) = 1 | − 1.56e5·2-s + 6.96e7·3-s + 1.57e10·4-s − 1.08e13·6-s + 1.55e14·7-s − 1.12e15·8-s − 7.11e14·9-s + 4.13e16·11-s + 1.09e18·12-s + 2.38e18·13-s − 2.42e19·14-s + 3.93e19·16-s + 2.91e20·17-s + 1.11e20·18-s − 3.61e20·19-s + 1.08e22·21-s − 6.44e21·22-s − 3.81e22·23-s − 7.79e22·24-s − 3.71e23·26-s − 4.36e23·27-s + 2.44e24·28-s + 1.65e24·29-s − 2.76e24·31-s + 3.47e24·32-s + 2.87e24·33-s − 4.55e25·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.933·3-s + 1.83·4-s − 1.57·6-s + 1.76·7-s − 1.40·8-s − 0.128·9-s + 0.271·11-s + 1.71·12-s + 0.993·13-s − 2.97·14-s + 0.533·16-s + 1.45·17-s + 0.215·18-s − 0.287·19-s + 1.64·21-s − 0.456·22-s − 1.29·23-s − 1.31·24-s − 1.67·26-s − 1.05·27-s + 3.23·28-s + 1.22·29-s − 0.682·31-s + 0.508·32-s + 0.253·33-s − 2.44·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(2.109746971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.109746971\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.56e5T + 8.58e9T^{2} \) |
| 3 | \( 1 - 6.96e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.55e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 4.13e16T + 2.32e34T^{2} \) |
| 13 | \( 1 - 2.38e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 2.91e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 3.61e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 3.81e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 1.65e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 2.76e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 8.70e24T + 5.63e51T^{2} \) |
| 41 | \( 1 - 2.69e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.33e26T + 8.02e53T^{2} \) |
| 47 | \( 1 - 2.88e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 2.92e28T + 7.96e56T^{2} \) |
| 59 | \( 1 - 1.51e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 2.88e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 8.72e29T + 1.82e60T^{2} \) |
| 71 | \( 1 - 3.08e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 2.41e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 3.21e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 7.79e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 1.41e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 7.13e32T + 3.65e65T^{2} \) |
| 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
−10.98918033164559113717302267640, −9.898256395148683921453402883598, −8.571018682446674074741428845991, −8.294892569036880700531852976283, −7.40178211661818774032456396171, −5.72402454593032957905694041752, −3.97265825954543872550102687665, −2.47663270076803029712408992125, −1.59681629857627453289229772195, −0.838717242861969204735569127414,
0.838717242861969204735569127414, 1.59681629857627453289229772195, 2.47663270076803029712408992125, 3.97265825954543872550102687665, 5.72402454593032957905694041752, 7.40178211661818774032456396171, 8.294892569036880700531852976283, 8.571018682446674074741428845991, 9.898256395148683921453402883598, 10.98918033164559113717302267640
