| L(s) = 1 | − 27·3-s − 160.·5-s + 729·9-s + 3.16e3·11-s − 4.77e3·13-s + 4.32e3·15-s − 1.30e4·17-s + 4.41e4·19-s − 6.50e4·23-s − 5.24e4·25-s − 1.96e4·27-s − 2.46e5·29-s + 3.00e5·31-s − 8.54e4·33-s + 5.16e5·37-s + 1.28e5·39-s − 3.77e5·41-s − 7.14e4·43-s − 1.16e5·45-s + 1.11e6·47-s + 3.51e5·51-s − 3.68e5·53-s − 5.06e5·55-s − 1.19e6·57-s + 1.03e6·59-s − 3.22e5·61-s + 7.64e5·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.573·5-s + 0.333·9-s + 0.716·11-s − 0.602·13-s + 0.330·15-s − 0.642·17-s + 1.47·19-s − 1.11·23-s − 0.671·25-s − 0.192·27-s − 1.87·29-s + 1.81·31-s − 0.413·33-s + 1.67·37-s + 0.347·39-s − 0.856·41-s − 0.136·43-s − 0.191·45-s + 1.56·47-s + 0.370·51-s − 0.340·53-s − 0.410·55-s − 0.851·57-s + 0.654·59-s − 0.182·61-s + 0.345·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 160.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 3.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.77e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.41e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 6.50e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.46e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.00e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.16e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.77e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.14e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.11e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.68e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.35e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.84e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.73e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.58e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.62e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.51e5T + 8.07e13T^{2} \) |
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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
−9.351216547489543125712962857641, −8.054589051994932707914925931460, −7.38958294946502525000641974281, −6.40823831291849144607518351871, −5.51094056034148444138109923426, −4.43541368917953031866705715585, −3.66245487501254801123571137257, −2.28214029114981452941124919665, −1.01808585803640046968074364387, 0,
1.01808585803640046968074364387, 2.28214029114981452941124919665, 3.66245487501254801123571137257, 4.43541368917953031866705715585, 5.51094056034148444138109923426, 6.40823831291849144607518351871, 7.38958294946502525000641974281, 8.054589051994932707914925931460, 9.351216547489543125712962857641
