| L(s) = 1 | − 70.0·2-s − 243·3-s + 2.86e3·4-s + 3.42e3·5-s + 1.70e4·6-s − 1.94e4·7-s − 5.72e4·8-s + 5.90e4·9-s − 2.40e5·10-s − 6.62e5·11-s − 6.96e5·12-s + 1.45e6·13-s + 1.36e6·14-s − 8.32e5·15-s − 1.85e6·16-s + 1.01e7·17-s − 4.13e6·18-s + 4.12e6·19-s + 9.81e6·20-s + 4.71e6·21-s + 4.64e7·22-s − 8.18e6·23-s + 1.39e7·24-s − 3.70e7·25-s − 1.01e8·26-s − 1.43e7·27-s − 5.56e7·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.577·3-s + 1.39·4-s + 0.490·5-s + 0.894·6-s − 0.436·7-s − 0.617·8-s + 0.333·9-s − 0.759·10-s − 1.24·11-s − 0.807·12-s + 1.08·13-s + 0.676·14-s − 0.283·15-s − 0.442·16-s + 1.72·17-s − 0.516·18-s + 0.382·19-s + 0.685·20-s + 0.252·21-s + 1.92·22-s − 0.265·23-s + 0.356·24-s − 0.759·25-s − 1.68·26-s − 0.192·27-s − 0.610·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 243T \) |
| 59 | \( 1 - 7.14e8T \) |
| good | 2 | \( 1 + 70.0T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.42e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.94e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.62e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.45e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.01e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.12e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 8.18e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.12e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 6.52e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.08e6T + 1.77e17T^{2} \) |
| 41 | \( 1 + 6.32e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 9.00e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.96e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 5.33e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 6.94e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.15e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.26e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.33e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.13e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.96e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.45e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.67e10T + 7.15e21T^{2} \) |
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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
−10.07775718870683467490692265903, −9.385925422100004290662182469009, −8.139883860208817536766749975529, −7.43895549834536392088276739581, −6.16420572572252219455860738555, −5.31953775534794197296243646193, −3.45004267569534614646321298655, −1.99231892289855929773363003208, −0.988560764940955036332289104997, 0,
0.988560764940955036332289104997, 1.99231892289855929773363003208, 3.45004267569534614646321298655, 5.31953775534794197296243646193, 6.16420572572252219455860738555, 7.43895549834536392088276739581, 8.139883860208817536766749975529, 9.385925422100004290662182469009, 10.07775718870683467490692265903
