| L(s) = 1 | + 4.04e4·2-s − 1.28e8·3-s − 6.95e9·4-s − 5.19e12·6-s − 4.50e13·7-s − 6.28e14·8-s + 1.09e16·9-s + 7.97e16·11-s + 8.94e17·12-s − 2.76e18·13-s − 1.81e18·14-s + 3.43e19·16-s − 5.61e19·17-s + 4.43e20·18-s + 1.90e21·19-s + 5.78e21·21-s + 3.22e21·22-s − 4.97e22·23-s + 8.07e22·24-s − 1.11e23·26-s − 6.95e23·27-s + 3.13e23·28-s − 1.05e24·29-s − 5.13e24·31-s + 6.78e24·32-s − 1.02e25·33-s − 2.26e24·34-s + ⋯ |
| L(s) = 1 | + 0.435·2-s − 1.72·3-s − 0.809·4-s − 0.751·6-s − 0.512·7-s − 0.789·8-s + 1.97·9-s + 0.523·11-s + 1.39·12-s − 1.15·13-s − 0.223·14-s + 0.465·16-s − 0.279·17-s + 0.859·18-s + 1.51·19-s + 0.882·21-s + 0.228·22-s − 1.69·23-s + 1.36·24-s − 0.501·26-s − 1.67·27-s + 0.414·28-s − 0.783·29-s − 1.26·31-s + 0.992·32-s − 0.902·33-s − 0.121·34-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
| \(L(17)\) |
\(\approx\) |
\(0.001815568773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001815568773\) |
| \(L(\frac{35}{2})\) |
|
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 | \( 1 - 4.04e4T + 8.58e9T^{2} \) |
| 3 | \( 1 + 1.28e8T + 5.55e15T^{2} \) |
| 7 | \( 1 + 4.50e13T + 7.73e27T^{2} \) |
| 11 | \( 1 - 7.97e16T + 2.32e34T^{2} \) |
| 13 | \( 1 + 2.76e18T + 5.75e36T^{2} \) |
| 17 | \( 1 + 5.61e19T + 4.02e40T^{2} \) |
| 19 | \( 1 - 1.90e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 4.97e22T + 8.65e44T^{2} \) |
| 29 | \( 1 + 1.05e24T + 1.81e48T^{2} \) |
| 31 | \( 1 + 5.13e24T + 1.64e49T^{2} \) |
| 37 | \( 1 + 1.64e25T + 5.63e51T^{2} \) |
| 41 | \( 1 + 3.83e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 6.26e25T + 8.02e53T^{2} \) |
| 47 | \( 1 + 5.32e27T + 1.51e55T^{2} \) |
| 53 | \( 1 - 8.17e27T + 7.96e56T^{2} \) |
| 59 | \( 1 + 2.46e29T + 2.74e58T^{2} \) |
| 61 | \( 1 + 5.26e28T + 8.23e58T^{2} \) |
| 67 | \( 1 + 2.38e30T + 1.82e60T^{2} \) |
| 71 | \( 1 - 2.98e30T + 1.23e61T^{2} \) |
| 73 | \( 1 + 8.89e30T + 3.08e61T^{2} \) |
| 79 | \( 1 + 2.50e31T + 4.18e62T^{2} \) |
| 83 | \( 1 - 2.29e31T + 2.13e63T^{2} \) |
| 89 | \( 1 - 6.67e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 6.85e32T + 3.65e65T^{2} \) |
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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
−11.66336547730618963128055804174, −10.14646637726083840418012884871, −9.390395284975864359596442272374, −7.45788323130559536294242991114, −6.23817223441638604806333759286, −5.40607987686942144851286394071, −4.58217113367459841650707592392, −3.45387404389218606159461053963, −1.51020020114823064013255013395, −0.01955221107091341903629084914,
0.01955221107091341903629084914, 1.51020020114823064013255013395, 3.45387404389218606159461053963, 4.58217113367459841650707592392, 5.40607987686942144851286394071, 6.23817223441638604806333759286, 7.45788323130559536294242991114, 9.390395284975864359596442272374, 10.14646637726083840418012884871, 11.66336547730618963128055804174
