| L(s) = 1 | + 1.65e5·2-s − 7.09e9·4-s + 2.05e12·5-s − 1.25e14·7-s − 6.84e15·8-s + 3.38e17·10-s + 1.70e18·11-s − 4.94e19·13-s − 2.06e19·14-s − 8.86e20·16-s − 1.32e21·17-s + 3.94e21·19-s − 1.45e22·20-s + 2.82e23·22-s + 3.48e23·23-s + 1.29e24·25-s − 8.17e24·26-s + 8.88e23·28-s − 3.21e25·29-s + 3.41e25·31-s + 8.88e25·32-s − 2.18e26·34-s − 2.56e26·35-s − 4.03e27·37-s + 6.51e26·38-s − 1.40e28·40-s − 8.65e27·41-s + ⋯ |
| L(s) = 1 | + 0.890·2-s − 0.206·4-s + 1.20·5-s − 0.203·7-s − 1.07·8-s + 1.07·10-s + 1.01·11-s − 1.58·13-s − 0.181·14-s − 0.750·16-s − 0.387·17-s + 0.165·19-s − 0.248·20-s + 0.907·22-s + 0.515·23-s + 0.444·25-s − 1.41·26-s + 0.0420·28-s − 0.822·29-s + 0.271·31-s + 0.406·32-s − 0.344·34-s − 0.244·35-s − 1.45·37-s + 0.147·38-s − 1.29·40-s − 0.517·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(18)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.65e5T + 3.43e10T^{2} \) |
| 5 | \( 1 - 2.05e12T + 2.91e24T^{2} \) |
| 7 | \( 1 + 1.25e14T + 3.78e29T^{2} \) |
| 11 | \( 1 - 1.70e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 4.94e19T + 9.72e38T^{2} \) |
| 17 | \( 1 + 1.32e21T + 1.16e43T^{2} \) |
| 19 | \( 1 - 3.94e21T + 5.70e44T^{2} \) |
| 23 | \( 1 - 3.48e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 3.21e25T + 1.52e51T^{2} \) |
| 31 | \( 1 - 3.41e25T + 1.57e52T^{2} \) |
| 37 | \( 1 + 4.03e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 8.65e27T + 2.80e56T^{2} \) |
| 43 | \( 1 - 9.89e26T + 1.48e57T^{2} \) |
| 47 | \( 1 + 1.95e29T + 3.33e58T^{2} \) |
| 53 | \( 1 - 9.96e29T + 2.23e60T^{2} \) |
| 59 | \( 1 + 3.91e30T + 9.54e61T^{2} \) |
| 61 | \( 1 - 7.64e29T + 3.06e62T^{2} \) |
| 67 | \( 1 + 1.64e32T + 8.17e63T^{2} \) |
| 71 | \( 1 + 7.65e31T + 6.22e64T^{2} \) |
| 73 | \( 1 + 7.08e32T + 1.64e65T^{2} \) |
| 79 | \( 1 - 2.34e33T + 2.61e66T^{2} \) |
| 83 | \( 1 + 5.18e33T + 1.47e67T^{2} \) |
| 89 | \( 1 + 1.44e34T + 1.69e68T^{2} \) |
| 97 | \( 1 + 2.87e34T + 3.44e69T^{2} \) |
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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
−13.06197772003827130530389826917, −11.90588328791562674017724079251, −9.908460954465951970801763210188, −9.042383222059042814522067099825, −6.83411473268576533357143528985, −5.63817878534075042379368075795, −4.58863712866837857355916202423, −3.09492109350285549218418395456, −1.77821526441288828818961979793, 0,
1.77821526441288828818961979793, 3.09492109350285549218418395456, 4.58863712866837857355916202423, 5.63817878534075042379368075795, 6.83411473268576533357143528985, 9.042383222059042814522067099825, 9.908460954465951970801763210188, 11.90588328791562674017724079251, 13.06197772003827130530389826917
