| L(s) = 1 | − 1.02e3·2-s + 1.56e5·3-s + 1.04e6·4-s − 1.60e8·6-s − 4.70e7·7-s − 1.07e9·8-s + 1.40e10·9-s − 7.06e10·11-s + 1.64e11·12-s − 4.22e11·13-s + 4.81e10·14-s + 1.09e12·16-s + 9.39e12·17-s − 1.44e13·18-s − 1.75e13·19-s − 7.37e12·21-s + 7.23e13·22-s − 2.12e13·23-s − 1.68e14·24-s + 4.32e14·26-s + 5.68e14·27-s − 4.93e13·28-s + 3.71e15·29-s + 4.95e15·31-s − 1.12e15·32-s − 1.10e16·33-s − 9.62e15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 0.5·4-s − 1.08·6-s − 0.0629·7-s − 0.353·8-s + 1.34·9-s − 0.821·11-s + 0.766·12-s − 0.849·13-s + 0.0445·14-s + 0.250·16-s + 1.13·17-s − 0.952·18-s − 0.657·19-s − 0.0964·21-s + 0.580·22-s − 0.106·23-s − 0.541·24-s + 0.600·26-s + 0.531·27-s − 0.0314·28-s + 1.63·29-s + 1.08·31-s − 0.176·32-s − 1.25·33-s − 0.799·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.02e3T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.56e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 4.70e7T + 5.58e17T^{2} \) |
| 11 | \( 1 + 7.06e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 4.22e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.39e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.75e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.12e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.71e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.95e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 2.52e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.20e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.31e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 6.43e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 3.87e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 6.13e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 9.11e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.33e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 4.55e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.93e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.55e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.70e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.83e19T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.19e21T + 5.27e41T^{2} \) |
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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
−10.26975999712600125515900031242, −9.690597603707867753355766770219, −8.346769066769793840433280836424, −7.941534910388565926077958438037, −6.66515051395736423201916314307, −4.87277969914629137265234236255, −3.26948256509761780771398708404, −2.57060912598023625161112233154, −1.47661445193045459318255441358, 0,
1.47661445193045459318255441358, 2.57060912598023625161112233154, 3.26948256509761780771398708404, 4.87277969914629137265234236255, 6.66515051395736423201916314307, 7.941534910388565926077958438037, 8.346769066769793840433280836424, 9.690597603707867753355766770219, 10.26975999712600125515900031242
