| L(s) = 1 | − 1.49e5·3-s − 3.99e7·5-s + 6.55e8·7-s + 1.18e10·9-s + 1.48e11·11-s − 4.89e11·13-s + 5.97e12·15-s + 7.60e12·17-s + 2.27e13·19-s − 9.78e13·21-s + 1.02e14·23-s + 1.12e15·25-s − 2.06e14·27-s + 1.03e15·29-s + 7.83e15·31-s − 2.21e16·33-s − 2.61e16·35-s − 3.12e16·37-s + 7.30e16·39-s − 7.59e16·41-s − 3.86e16·43-s − 4.73e17·45-s + 9.72e15·47-s − 1.29e17·49-s − 1.13e18·51-s + 7.40e16·53-s − 5.92e18·55-s + ⋯ |
| L(s) = 1 | − 1.46·3-s − 1.83·5-s + 0.876·7-s + 1.13·9-s + 1.72·11-s − 0.984·13-s + 2.67·15-s + 0.915·17-s + 0.851·19-s − 1.27·21-s + 0.514·23-s + 2.35·25-s − 0.193·27-s + 0.456·29-s + 1.71·31-s − 2.51·33-s − 1.60·35-s − 1.06·37-s + 1.43·39-s − 0.883·41-s − 0.272·43-s − 2.07·45-s + 0.0269·47-s − 0.231·49-s − 1.33·51-s + 0.0581·53-s − 3.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.118131146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.118131146\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.49e5T + 1.04e10T^{2} \) |
| 5 | \( 1 + 3.99e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 6.55e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.48e11T + 7.40e21T^{2} \) |
| 13 | \( 1 + 4.89e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 7.60e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.27e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.02e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.03e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 7.83e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.12e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 7.59e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 3.86e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 9.72e15T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.40e16T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.92e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.19e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.58e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 5.10e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 5.30e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 6.52e17T + 7.08e39T^{2} \) |
| 83 | \( 1 - 8.47e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.36e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.65e20T + 5.27e41T^{2} \) |
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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
−11.44869697776677355274363464570, −10.16380855820824913023019402471, −8.549358334344023996660037960302, −7.43638141808381010213885274146, −6.63913163879393071455460503994, −5.11272024428774207213744436899, −4.46804182487959774385925990853, −3.36219319555456284569846310234, −1.22621594533626805506398369273, −0.59373533875983934041675304947,
0.59373533875983934041675304947, 1.22621594533626805506398369273, 3.36219319555456284569846310234, 4.46804182487959774385925990853, 5.11272024428774207213744436899, 6.63913163879393071455460503994, 7.43638141808381010213885274146, 8.549358334344023996660037960302, 10.16380855820824913023019402471, 11.44869697776677355274363464570
