| L(s) = 1 | + 113.·2-s − 729·3-s + 4.65e3·4-s − 1.36e4·5-s − 8.26e4·6-s − 3.26e4·7-s − 4.00e5·8-s + 5.31e5·9-s − 1.54e6·10-s − 9.33e6·11-s − 3.39e6·12-s + 2.15e7·13-s − 3.69e6·14-s + 9.95e6·15-s − 8.35e7·16-s − 8.77e6·17-s + 6.02e7·18-s + 8.65e7·19-s − 6.36e7·20-s + 2.37e7·21-s − 1.05e9·22-s − 1.33e8·23-s + 2.92e8·24-s − 1.03e9·25-s + 2.44e9·26-s − 3.87e8·27-s − 1.51e8·28-s + ⋯ |
| L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.568·4-s − 0.390·5-s − 0.723·6-s − 0.104·7-s − 0.540·8-s + 0.333·9-s − 0.489·10-s − 1.58·11-s − 0.328·12-s + 1.23·13-s − 0.131·14-s + 0.225·15-s − 1.24·16-s − 0.0881·17-s + 0.417·18-s + 0.421·19-s − 0.222·20-s + 0.0604·21-s − 1.98·22-s − 0.187·23-s + 0.312·24-s − 0.847·25-s + 1.55·26-s − 0.192·27-s − 0.0595·28-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
| \(L(7)\) |
\(\approx\) |
\(1.595359577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.595359577\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 - 113.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 1.36e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 3.26e4T + 9.68e10T^{2} \) |
| 11 | \( 1 + 9.33e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.15e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 8.77e6T + 9.90e15T^{2} \) |
| 19 | \( 1 - 8.65e7T + 4.20e16T^{2} \) |
| 23 | \( 1 + 1.33e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.35e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 7.90e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.32e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.73e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 7.82e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 9.20e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.70e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 6.91e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 2.36e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 4.03e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 6.40e11T + 1.67e24T^{2} \) |
| 79 | \( 1 + 9.80e9T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.21e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 1.78e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 8.80e12T + 6.73e25T^{2} \) |
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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
−10.80731437927768841793886920257, −9.443037945346993749240864165715, −8.153968594842662610130178027117, −7.05993815916319641527841796638, −5.74604489389505278175469962237, −5.37709478636605815078642419249, −4.10795781696948108951889584510, −3.37212061393029542798787679658, −2.07372780578904509419853346913, −0.43345018413081161620748289648,
0.43345018413081161620748289648, 2.07372780578904509419853346913, 3.37212061393029542798787679658, 4.10795781696948108951889584510, 5.37709478636605815078642419249, 5.74604489389505278175469962237, 7.05993815916319641527841796638, 8.153968594842662610130178027117, 9.443037945346993749240864165715, 10.80731437927768841793886920257
