| L(s) = 1 | + 3.47e4·2-s − 4.98e7·3-s − 7.38e9·4-s − 1.73e12·6-s + 1.47e14·7-s − 5.55e14·8-s − 3.07e15·9-s + 5.98e16·11-s + 3.67e17·12-s + 1.87e18·13-s + 5.13e18·14-s + 4.41e19·16-s + 1.50e19·17-s − 1.06e20·18-s + 9.64e20·19-s − 7.36e21·21-s + 2.07e21·22-s − 2.94e22·23-s + 2.76e22·24-s + 6.50e22·26-s + 4.30e23·27-s − 1.09e24·28-s + 2.20e24·29-s + 2.16e24·31-s + 6.30e24·32-s − 2.97e24·33-s + 5.21e23·34-s + ⋯ |
| L(s) = 1 | + 0.374·2-s − 0.668·3-s − 0.859·4-s − 0.250·6-s + 1.68·7-s − 0.697·8-s − 0.553·9-s + 0.392·11-s + 0.574·12-s + 0.780·13-s + 0.630·14-s + 0.598·16-s + 0.0748·17-s − 0.207·18-s + 0.767·19-s − 1.12·21-s + 0.147·22-s − 1.00·23-s + 0.465·24-s + 0.292·26-s + 1.03·27-s − 1.44·28-s + 1.63·29-s + 0.533·31-s + 0.921·32-s − 0.262·33-s + 0.0280·34-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
| \(L(17)\) |
\(\approx\) |
\(1.970363266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970363266\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 - 3.47e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 4.98e7T + 5.55e15T^{2} \) |
| 7 | \( 1 - 1.47e14T + 7.73e27T^{2} \) |
| 11 | \( 1 - 5.98e16T + 2.32e34T^{2} \) |
| 13 | \( 1 - 1.87e18T + 5.75e36T^{2} \) |
| 17 | \( 1 - 1.50e19T + 4.02e40T^{2} \) |
| 19 | \( 1 - 9.64e20T + 1.58e42T^{2} \) |
| 23 | \( 1 + 2.94e22T + 8.65e44T^{2} \) |
| 29 | \( 1 - 2.20e24T + 1.81e48T^{2} \) |
| 31 | \( 1 - 2.16e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.14e26T + 5.63e51T^{2} \) |
| 41 | \( 1 + 5.38e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 5.52e25T + 8.02e53T^{2} \) |
| 47 | \( 1 + 2.40e26T + 1.51e55T^{2} \) |
| 53 | \( 1 + 3.29e28T + 7.96e56T^{2} \) |
| 59 | \( 1 + 2.15e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 2.06e29T + 8.23e58T^{2} \) |
| 67 | \( 1 - 2.34e30T + 1.82e60T^{2} \) |
| 71 | \( 1 - 5.77e29T + 1.23e61T^{2} \) |
| 73 | \( 1 + 1.07e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.26e31T + 4.18e62T^{2} \) |
| 83 | \( 1 + 2.41e30T + 2.13e63T^{2} \) |
| 89 | \( 1 - 2.76e32T + 2.13e64T^{2} \) |
| 97 | \( 1 + 1.55e32T + 3.65e65T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.53833302511426996215625393302, −10.33897877188424693662589563506, −8.752299098961472069115900221764, −8.065539463872334395651759021154, −6.27179569494711098276888669203, −5.22453504490289051286462819561, −4.58319974743169356385166367517, −3.30873776780204431012978048170, −1.59670858923265381363517782888, −0.63777124357625994695377815527,
0.63777124357625994695377815527, 1.59670858923265381363517782888, 3.30873776780204431012978048170, 4.58319974743169356385166367517, 5.22453504490289051286462819561, 6.27179569494711098276888669203, 8.065539463872334395651759021154, 8.752299098961472069115900221764, 10.33897877188424693662589563506, 11.53833302511426996215625393302
