| L(s) = 1 | − 1.65e5·2-s − 3.45e8·3-s − 7.09e9·4-s − 2.05e12·5-s + 5.71e13·6-s − 1.25e14·7-s + 6.84e15·8-s + 6.96e16·9-s + 3.38e17·10-s − 1.70e18·11-s + 2.45e18·12-s − 4.94e19·13-s + 2.06e19·14-s + 7.09e20·15-s − 8.86e20·16-s + 1.32e21·17-s − 1.14e22·18-s + 3.94e21·19-s + 1.45e22·20-s + 4.32e22·21-s + 2.82e23·22-s − 3.48e23·23-s − 2.36e24·24-s + 1.29e24·25-s + 8.17e24·26-s − 6.77e24·27-s + 8.88e23·28-s + ⋯ |
| L(s) = 1 | − 0.890·2-s − 1.54·3-s − 0.206·4-s − 1.20·5-s + 1.37·6-s − 0.203·7-s + 1.07·8-s + 1.39·9-s + 1.07·10-s − 1.01·11-s + 0.319·12-s − 1.58·13-s + 0.181·14-s + 1.85·15-s − 0.750·16-s + 0.387·17-s − 1.23·18-s + 0.165·19-s + 0.248·20-s + 0.314·21-s + 0.907·22-s − 0.515·23-s − 1.66·24-s + 0.444·25-s + 1.41·26-s − 0.605·27-s + 0.0420·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(36-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+35/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(18)\) |
\(\approx\) |
\(0.1353440882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1353440882\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 1.65e5T + 3.43e10T^{2} \) |
| 3 | \( 1 + 3.45e8T + 5.00e16T^{2} \) |
| 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
−23.67785201511373650820667889069, −22.44595811460289251393713057497, −19.22014790230308983267427916080, −17.59830144349165301023594214509, −16.11351508686302067000630521883, −12.14124545628775680868267838282, −10.34826311131646356683386295159, −7.56742165052681597633141970493, −4.84357168970594102502750076522, −0.37295365828732270267756485604,
0.37295365828732270267756485604, 4.84357168970594102502750076522, 7.56742165052681597633141970493, 10.34826311131646356683386295159, 12.14124545628775680868267838282, 16.11351508686302067000630521883, 17.59830144349165301023594214509, 19.22014790230308983267427916080, 22.44595811460289251393713057497, 23.67785201511373650820667889069
