| L(s) = 1 | + 174.·2-s + 729·3-s + 2.23e4·4-s − 3.85e4·5-s + 1.27e5·6-s − 2.36e5·7-s + 2.47e6·8-s + 5.31e5·9-s − 6.73e6·10-s + 7.91e6·11-s + 1.62e7·12-s + 3.86e6·13-s − 4.12e7·14-s − 2.81e7·15-s + 2.49e8·16-s − 8.38e7·17-s + 9.28e7·18-s + 1.09e8·19-s − 8.61e8·20-s − 1.72e8·21-s + 1.38e9·22-s + 5.43e8·23-s + 1.80e9·24-s + 2.66e8·25-s + 6.75e8·26-s + 3.87e8·27-s − 5.27e9·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s + 0.577·3-s + 2.72·4-s − 1.10·5-s + 1.11·6-s − 0.758·7-s + 3.33·8-s + 0.333·9-s − 2.13·10-s + 1.34·11-s + 1.57·12-s + 0.222·13-s − 1.46·14-s − 0.637·15-s + 3.71·16-s − 0.842·17-s + 0.643·18-s + 0.535·19-s − 3.01·20-s − 0.437·21-s + 2.59·22-s + 0.765·23-s + 1.92·24-s + 0.218·25-s + 0.428·26-s + 0.192·27-s − 2.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(10.10255946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.10255946\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
| good | 2 | \( 1 - 174.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 3.85e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.36e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 7.91e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 3.86e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.38e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.09e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 5.43e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 6.42e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.97e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.07e9T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.58e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.08e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.45e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 8.09e10T + 2.60e22T^{2} \) |
| 61 | \( 1 - 8.52e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 6.87e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.01e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.75e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.74e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.86e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 7.08e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.48e12T + 6.73e25T^{2} \) |
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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
−10.87664228457222571408676394297, −9.372751444931284540988421735793, −7.957313189235232707444149213816, −6.89616478289069836162339689998, −6.32264210342629343333440497153, −4.83084000343980337324173575923, −3.91160997281127374968141324993, −3.44331924046032676377382667392, −2.40053383700012406537476344311, −1.01358434183940714279543365911,
1.01358434183940714279543365911, 2.40053383700012406537476344311, 3.44331924046032676377382667392, 3.91160997281127374968141324993, 4.83084000343980337324173575923, 6.32264210342629343333440497153, 6.89616478289069836162339689998, 7.957313189235232707444149213816, 9.372751444931284540988421735793, 10.87664228457222571408676394297
