| L(s) = 1 | − 4.96e4·2-s + 1.55e7·3-s − 6.12e9·4-s − 7.71e11·6-s + 1.22e14·7-s + 7.30e14·8-s − 5.31e15·9-s + 2.15e17·11-s − 9.50e16·12-s + 1.07e18·13-s − 6.06e18·14-s + 1.62e19·16-s − 2.54e20·17-s + 2.64e20·18-s − 1.22e21·19-s + 1.89e21·21-s − 1.07e22·22-s + 5.11e21·23-s + 1.13e22·24-s − 5.35e22·26-s − 1.68e23·27-s − 7.47e23·28-s − 1.64e23·29-s − 6.75e24·31-s − 7.08e24·32-s + 3.34e24·33-s + 1.26e25·34-s + ⋯ |
| L(s) = 1 | − 0.536·2-s + 0.208·3-s − 0.712·4-s − 0.111·6-s + 1.38·7-s + 0.918·8-s − 0.956·9-s + 1.41·11-s − 0.148·12-s + 0.449·13-s − 0.744·14-s + 0.220·16-s − 1.26·17-s + 0.512·18-s − 0.970·19-s + 0.289·21-s − 0.758·22-s + 0.173·23-s + 0.191·24-s − 0.240·26-s − 0.407·27-s − 0.990·28-s − 0.122·29-s − 1.66·31-s − 1.03·32-s + 0.294·33-s + 0.680·34-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
| \(L(17)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 + 4.96e4T + 8.58e9T^{2} \) |
| 3 | \( 1 - 1.55e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.22e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 2.15e17T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.07e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 2.54e20T + 4.02e40T^{2} \) |
| 19 | \( 1 + 1.22e21T + 1.58e42T^{2} \) |
| 23 | \( 1 - 5.11e21T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.64e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 6.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 7.46e25T + 5.63e51T^{2} \) |
| 41 | \( 1 - 4.96e26T + 1.66e53T^{2} \) |
| 43 | \( 1 - 1.99e26T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.16e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 3.60e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 1.87e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.18e27T + 8.23e58T^{2} \) |
| 67 | \( 1 + 4.85e29T + 1.82e60T^{2} \) |
| 71 | \( 1 + 3.42e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 7.01e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.95e30T + 4.18e62T^{2} \) |
| 83 | \( 1 - 1.23e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 7.05e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 7.71e32T + 3.65e65T^{2} \) |
| 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
−10.82066407522207181478719058228, −9.024962115344064443715767243591, −8.759849456726308000212499655071, −7.58861942873726912676026015307, −6.03485395378650921646745379754, −4.67357934871861206024189243832, −3.87978255799146502701596202517, −2.10616539166985670501968963020, −1.19107452888541694831127487855, 0,
1.19107452888541694831127487855, 2.10616539166985670501968963020, 3.87978255799146502701596202517, 4.67357934871861206024189243832, 6.03485395378650921646745379754, 7.58861942873726912676026015307, 8.759849456726308000212499655071, 9.024962115344064443715767243591, 10.82066407522207181478719058228
