| L(s) = 1 | − 238.·2-s + 5.90e4·3-s − 2.04e6·4-s − 1.41e7·6-s − 6.22e8·7-s + 9.88e8·8-s + 3.48e9·9-s + 1.17e10·11-s − 1.20e11·12-s − 6.64e11·13-s + 1.48e11·14-s + 4.04e12·16-s − 1.24e13·17-s − 8.33e11·18-s + 1.38e13·19-s − 3.67e13·21-s − 2.81e12·22-s − 2.44e14·23-s + 5.83e13·24-s + 1.58e14·26-s + 2.05e14·27-s + 1.27e15·28-s − 2.45e15·29-s + 6.56e15·31-s − 3.03e15·32-s + 6.94e14·33-s + 2.96e15·34-s + ⋯ |
| L(s) = 1 | − 0.164·2-s + 0.577·3-s − 0.972·4-s − 0.0952·6-s − 0.833·7-s + 0.325·8-s + 0.333·9-s + 0.136·11-s − 0.561·12-s − 1.33·13-s + 0.137·14-s + 0.919·16-s − 1.49·17-s − 0.0549·18-s + 0.519·19-s − 0.481·21-s − 0.0225·22-s − 1.23·23-s + 0.187·24-s + 0.220·26-s + 0.192·27-s + 0.810·28-s − 1.08·29-s + 1.43·31-s − 0.477·32-s + 0.0789·33-s + 0.246·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(0.5217492095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5217492095\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 238.T + 2.09e6T^{2} \) |
| 7 | \( 1 + 6.22e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.17e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 6.64e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.24e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 1.38e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.44e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 2.45e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.56e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 8.38e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.59e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.03e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 6.98e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.25e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.40e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 6.88e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.90e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 6.30e18T + 7.52e38T^{2} \) |
| 73 | \( 1 - 3.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 9.29e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 5.72e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.21e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 7.43e20T + 5.27e41T^{2} \) |
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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
−10.12818156360603474438161321307, −9.592523855116571290606980244443, −8.656992263777010985753797454903, −7.60729599798571530924031557441, −6.44030109700605612739127073478, −4.98691442469108443176304095383, −4.07312482172247712414783472154, −2.98315693858818879666609301467, −1.81826420034467378694596775445, −0.28940563458108484476977778477,
0.28940563458108484476977778477, 1.81826420034467378694596775445, 2.98315693858818879666609301467, 4.07312482172247712414783472154, 4.98691442469108443176304095383, 6.44030109700605612739127073478, 7.60729599798571530924031557441, 8.656992263777010985753797454903, 9.592523855116571290606980244443, 10.12818156360603474438161321307
