| L(s) = 1 | − 5.90e4·3-s − 3.08e7·5-s − 1.10e9·7-s + 3.48e9·9-s + 1.54e11·11-s + 5.78e11·13-s + 1.82e12·15-s + 1.59e12·17-s + 7.29e12·19-s + 6.53e13·21-s − 1.56e14·23-s + 4.73e14·25-s − 2.05e14·27-s − 1.80e15·29-s + 5.98e15·31-s − 9.13e15·33-s + 3.40e16·35-s + 1.92e16·37-s − 3.41e16·39-s − 1.01e17·41-s − 2.34e17·43-s − 1.07e17·45-s + 1.31e17·47-s + 6.64e17·49-s − 9.43e16·51-s + 2.51e18·53-s − 4.76e18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.41·5-s − 1.47·7-s + 0.333·9-s + 1.79·11-s + 1.16·13-s + 0.815·15-s + 0.192·17-s + 0.272·19-s + 0.854·21-s − 0.786·23-s + 0.993·25-s − 0.192·27-s − 0.798·29-s + 1.31·31-s − 1.03·33-s + 2.08·35-s + 0.658·37-s − 0.672·39-s − 1.18·41-s − 1.65·43-s − 0.470·45-s + 0.364·47-s + 1.18·49-s − 0.110·51-s + 1.97·53-s − 2.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + 5.90e4T \) |
| good | 5 | \( 1 + 3.08e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 1.10e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.54e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 5.78e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.59e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 7.29e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.56e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.80e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.98e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.92e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.01e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.34e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.31e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 2.51e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.37e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.88e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.11e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.02e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.07e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.48e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 9.43e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.18e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.66e20T + 5.27e41T^{2} \) |
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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
−12.09344468568536851582970310246, −11.51447532228894117432890620804, −9.928240594890928701792752644719, −8.583146354264335176208896302346, −6.97647153336509637618572622222, −6.11706433279956963331425119331, −4.06636508542041307471171372788, −3.45902161990113917971209248954, −1.09689696765026389826204198252, 0,
1.09689696765026389826204198252, 3.45902161990113917971209248954, 4.06636508542041307471171372788, 6.11706433279956963331425119331, 6.97647153336509637618572622222, 8.583146354264335176208896302346, 9.928240594890928701792752644719, 11.51447532228894117432890620804, 12.09344468568536851582970310246
