| L(s) = 1 | − 36.7·2-s − 81·3-s + 840.·4-s + 2.30e3·5-s + 2.97e3·6-s − 1.50e3·7-s − 1.20e4·8-s + 6.56e3·9-s − 8.47e4·10-s + 3.54e4·11-s − 6.81e4·12-s − 1.17e5·13-s + 5.55e4·14-s − 1.86e5·15-s + 1.42e4·16-s + 4.66e5·17-s − 2.41e5·18-s − 9.13e4·19-s + 1.93e6·20-s + 1.22e5·21-s − 1.30e6·22-s + 5.07e5·23-s + 9.79e5·24-s + 3.36e6·25-s + 4.32e6·26-s − 5.31e5·27-s − 1.26e6·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.64·4-s + 1.64·5-s + 0.938·6-s − 0.237·7-s − 1.04·8-s + 0.333·9-s − 2.68·10-s + 0.730·11-s − 0.948·12-s − 1.14·13-s + 0.386·14-s − 0.952·15-s + 0.0543·16-s + 1.35·17-s − 0.541·18-s − 0.160·19-s + 2.70·20-s + 0.137·21-s − 1.18·22-s + 0.377·23-s + 0.602·24-s + 1.72·25-s + 1.85·26-s − 0.192·27-s − 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.166494532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.166494532\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 36.7T + 512T^{2} \) |
| 5 | \( 1 - 2.30e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.50e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.54e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.17e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 4.66e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.13e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.07e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.06e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.34e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.50e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.13e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 5.74e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.17e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.11e6T + 3.29e15T^{2} \) |
| 61 | \( 1 + 2.97e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.83e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.32e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.01e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.98e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.95e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.05e9T + 7.60e17T^{2} \) |
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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
−10.50947758691692415850235697748, −9.754954560647851486147624146745, −9.489793833032587523530920403071, −8.139724958291314311149640085063, −6.88985562961144276442043067158, −6.17781322505133278099662621396, −4.96565897976513819419421350565, −2.69978385182041224872904863915, −1.57942228328585620433987897974, −0.74950179864418787533749655292,
0.74950179864418787533749655292, 1.57942228328585620433987897974, 2.69978385182041224872904863915, 4.96565897976513819419421350565, 6.17781322505133278099662621396, 6.88985562961144276442043067158, 8.139724958291314311149640085063, 9.489793833032587523530920403071, 9.754954560647851486147624146745, 10.50947758691692415850235697748
