L(s) = 1 | + 149.·2-s + 729·3-s + 1.41e4·4-s + 6.31e4·5-s + 1.08e5·6-s + 1.47e5·7-s + 8.82e5·8-s + 5.31e5·9-s + 9.42e6·10-s + 7.64e5·11-s + 1.02e7·12-s + 1.16e7·13-s + 2.20e7·14-s + 4.60e7·15-s + 1.62e7·16-s − 1.70e8·17-s + 7.93e7·18-s − 2.91e7·19-s + 8.90e8·20-s + 1.07e8·21-s + 1.14e8·22-s + 2.00e7·23-s + 6.43e8·24-s + 2.76e9·25-s + 1.74e9·26-s + 3.87e8·27-s + 2.08e9·28-s + ⋯ |
L(s) = 1 | + 1.64·2-s + 0.577·3-s + 1.72·4-s + 1.80·5-s + 0.952·6-s + 0.473·7-s + 1.18·8-s + 0.333·9-s + 2.98·10-s + 0.130·11-s + 0.993·12-s + 0.671·13-s + 0.781·14-s + 1.04·15-s + 0.241·16-s − 1.70·17-s + 0.549·18-s − 0.142·19-s + 3.11·20-s + 0.273·21-s + 0.214·22-s + 0.0281·23-s + 0.687·24-s + 2.26·25-s + 1.10·26-s + 0.192·27-s + 0.815·28-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
\(L(7)\) |
\(\approx\) |
\(13.19577701\) |
\(L(\frac12)\) |
\(\approx\) |
\(13.19577701\) |
\(L(\frac{15}{2})\) |
|
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 | \( 1 - 149.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 6.31e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 1.47e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 7.64e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.16e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.70e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.91e7T + 4.20e16T^{2} \) |
| 23 | \( 1 - 2.00e7T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.66e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.20e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.76e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.43e9T + 9.25e20T^{2} \) |
| 43 | \( 1 + 3.32e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 4.72e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 5.80e10T + 2.60e22T^{2} \) |
| 61 | \( 1 + 2.95e10T + 1.61e23T^{2} \) |
| 67 | \( 1 + 9.30e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.05e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 2.06e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.54e12T + 4.66e24T^{2} \) |
| 83 | \( 1 - 2.83e12T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.27e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.78e12T + 6.73e25T^{2} \) |
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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
−10.51661904157336474693184326595, −9.397497557741108452563436898717, −8.395708571669314592388307981740, −6.57970695990001732572135796912, −6.27080788249483728908937707166, −5.02669001223948394006269485005, −4.32615578309732603720895603164, −2.87638438095437366792503183950, −2.24738513453781160546023267098, −1.31783663745694760513173757896,
1.31783663745694760513173757896, 2.24738513453781160546023267098, 2.87638438095437366792503183950, 4.32615578309732603720895603164, 5.02669001223948394006269485005, 6.27080788249483728908937707166, 6.57970695990001732572135796912, 8.395708571669314592388307981740, 9.397497557741108452563436898717, 10.51661904157336474693184326595