| L(s) = 1 | + 9.85e7·3-s − 2.11e12·5-s + 1.18e15·7-s − 4.03e16·9-s − 7.76e17·11-s − 2.10e18·13-s − 2.08e20·15-s + 4.70e21·17-s − 3.36e22·19-s + 1.16e23·21-s − 5.13e23·23-s + 1.55e24·25-s − 8.90e24·27-s − 2.68e25·29-s − 1.14e26·31-s − 7.64e25·33-s − 2.49e27·35-s − 3.17e27·37-s − 2.07e26·39-s − 2.95e28·41-s + 1.40e28·43-s + 8.51e28·45-s − 1.59e29·47-s + 1.01e30·49-s + 4.63e29·51-s − 7.01e29·53-s + 1.63e30·55-s + ⋯ |
| L(s) = 1 | + 0.440·3-s − 1.23·5-s + 1.91·7-s − 0.806·9-s − 0.463·11-s − 0.0674·13-s − 0.545·15-s + 1.37·17-s − 1.40·19-s + 0.844·21-s − 0.758·23-s + 0.533·25-s − 0.795·27-s − 0.686·29-s − 0.911·31-s − 0.203·33-s − 2.37·35-s − 1.14·37-s − 0.0297·39-s − 1.76·41-s + 0.365·43-s + 0.998·45-s − 0.871·47-s + 2.68·49-s + 0.606·51-s − 0.469·53-s + 0.573·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| good | 3 | \( 1 - 9.85e7T + 5.00e16T^{2} \) |
| 5 | \( 1 + 2.11e12T + 2.91e24T^{2} \) |
| 7 | \( 1 - 1.18e15T + 3.78e29T^{2} \) |
| 11 | \( 1 + 7.76e17T + 2.81e36T^{2} \) |
| 13 | \( 1 + 2.10e18T + 9.72e38T^{2} \) |
| 17 | \( 1 - 4.70e21T + 1.16e43T^{2} \) |
| 19 | \( 1 + 3.36e22T + 5.70e44T^{2} \) |
| 23 | \( 1 + 5.13e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 2.68e25T + 1.52e51T^{2} \) |
| 31 | \( 1 + 1.14e26T + 1.57e52T^{2} \) |
| 37 | \( 1 + 3.17e27T + 7.71e54T^{2} \) |
| 41 | \( 1 + 2.95e28T + 2.80e56T^{2} \) |
| 43 | \( 1 - 1.40e28T + 1.48e57T^{2} \) |
| 47 | \( 1 + 1.59e29T + 3.33e58T^{2} \) |
| 53 | \( 1 + 7.01e29T + 2.23e60T^{2} \) |
| 59 | \( 1 + 4.07e30T + 9.54e61T^{2} \) |
| 61 | \( 1 + 1.37e31T + 3.06e62T^{2} \) |
| 67 | \( 1 - 1.04e32T + 8.17e63T^{2} \) |
| 71 | \( 1 - 1.43e32T + 6.22e64T^{2} \) |
| 73 | \( 1 - 6.55e32T + 1.64e65T^{2} \) |
| 79 | \( 1 - 2.85e32T + 2.61e66T^{2} \) |
| 83 | \( 1 + 5.01e33T + 1.47e67T^{2} \) |
| 89 | \( 1 - 2.12e34T + 1.69e68T^{2} \) |
| 97 | \( 1 - 2.50e32T + 3.44e69T^{2} \) |
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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
−15.15775775512923627644159452633, −14.28315083837135092532429722256, −12.01450747588049854028099336865, −10.92751022972482810258196771346, −8.381518577231603154586730584182, −7.78130390252384120519825743408, −5.17626884591569885794689381459, −3.70063862090065451799685174861, −1.88037850481774459175859408422, 0,
1.88037850481774459175859408422, 3.70063862090065451799685174861, 5.17626884591569885794689381459, 7.78130390252384120519825743408, 8.381518577231603154586730584182, 10.92751022972482810258196771346, 12.01450747588049854028099336865, 14.28315083837135092532429722256, 15.15775775512923627644159452633
