| L(s) = 1 | − 2.70e3·2-s + 5.90e4·3-s + 5.22e6·4-s − 1.59e8·6-s + 9.90e7·7-s − 8.46e9·8-s + 3.48e9·9-s + 6.27e10·11-s + 3.08e11·12-s − 7.95e11·13-s − 2.68e11·14-s + 1.19e13·16-s − 1.22e13·17-s − 9.43e12·18-s − 1.18e13·19-s + 5.84e12·21-s − 1.69e14·22-s + 3.16e14·23-s − 4.99e14·24-s + 2.15e15·26-s + 2.05e14·27-s + 5.17e14·28-s − 9.42e14·29-s + 4.96e15·31-s − 1.45e16·32-s + 3.70e15·33-s + 3.32e16·34-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.577·3-s + 2.49·4-s − 1.07·6-s + 0.132·7-s − 2.78·8-s + 0.333·9-s + 0.729·11-s + 1.43·12-s − 1.60·13-s − 0.247·14-s + 2.71·16-s − 1.47·17-s − 0.622·18-s − 0.444·19-s + 0.0765·21-s − 1.36·22-s + 1.59·23-s − 1.60·24-s + 2.99·26-s + 0.192·27-s + 0.330·28-s − 0.415·29-s + 1.08·31-s − 2.28·32-s + 0.420·33-s + 2.76·34-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
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.70e3T + 2.09e6T^{2} \) |
| 7 | \( 1 - 9.90e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 6.27e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.95e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.22e13T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.18e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.16e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 9.42e14T + 5.13e30T^{2} \) |
| 31 | \( 1 - 4.96e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.95e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.21e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 4.99e15T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.66e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 5.56e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.92e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.85e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.68e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.71e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.95e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 6.84e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.44e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.66e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.10e20T + 5.27e41T^{2} \) |
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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
−9.691037649778627987640979336601, −9.146541877983922424382534310189, −8.199046788607312791595614903588, −7.21185544117247262516584061774, −6.48825924675870520656314409615, −4.57769832234045522850512226109, −2.82376789335312495363199229178, −2.12802134486928455323917729947, −1.03757547014612165849838908279, 0,
1.03757547014612165849838908279, 2.12802134486928455323917729947, 2.82376789335312495363199229178, 4.57769832234045522850512226109, 6.48825924675870520656314409615, 7.21185544117247262516584061774, 8.199046788607312791595614903588, 9.146541877983922424382534310189, 9.691037649778627987640979336601
