| L(s) = 1 | − 309.·5-s − 84.7·7-s − 7.75e3·11-s + 1.18e4·13-s + 8.28e3·17-s + 4.72e4·19-s + 5.27e4·23-s + 1.78e4·25-s − 5.72e4·29-s − 9.64e4·31-s + 2.62e4·35-s + 1.78e5·37-s + 1.27e5·41-s + 7.08e5·43-s − 8.72e5·47-s − 8.16e5·49-s − 3.64e5·53-s + 2.40e6·55-s + 2.07e6·59-s − 1.10e6·61-s − 3.66e6·65-s − 3.26e6·67-s + 2.68e6·71-s − 2.96e6·73-s + 6.57e5·77-s − 5.00e6·79-s − 1.73e6·83-s + ⋯ |
| L(s) = 1 | − 1.10·5-s − 0.0934·7-s − 1.75·11-s + 1.49·13-s + 0.408·17-s + 1.57·19-s + 0.904·23-s + 0.227·25-s − 0.436·29-s − 0.581·31-s + 0.103·35-s + 0.579·37-s + 0.288·41-s + 1.35·43-s − 1.22·47-s − 0.991·49-s − 0.336·53-s + 1.94·55-s + 1.31·59-s − 0.621·61-s − 1.65·65-s − 1.32·67-s + 0.889·71-s − 0.890·73-s + 0.164·77-s − 1.14·79-s − 0.333·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 309.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 84.7T + 8.23e5T^{2} \) |
| 11 | \( 1 + 7.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.18e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 8.28e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.72e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.27e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 5.72e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 9.64e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.78e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.27e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.08e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.72e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.64e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.07e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.10e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.26e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.68e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.96e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.73e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.55e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.78e7T + 8.07e13T^{2} \) |
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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
−10.03830659533617071864173611603, −8.862451956509985075405203309182, −7.85867700189667973723516570309, −7.41259188854844610271172004102, −5.91081692062322569337204931578, −4.97709329864992507640886536937, −3.68474771181837190189849573237, −2.89675894051725167270304122111, −1.15190826852163754412297011824, 0,
1.15190826852163754412297011824, 2.89675894051725167270304122111, 3.68474771181837190189849573237, 4.97709329864992507640886536937, 5.91081692062322569337204931578, 7.41259188854844610271172004102, 7.85867700189667973723516570309, 8.862451956509985075405203309182, 10.03830659533617071864173611603
