| L(s) = 1 | + 20.9·2-s − 243·3-s − 1.60e3·4-s + 5.79e3·5-s − 5.09e3·6-s + 2.21e4·7-s − 7.67e4·8-s + 5.90e4·9-s + 1.21e5·10-s + 9.36e5·11-s + 3.90e5·12-s − 1.11e6·13-s + 4.65e5·14-s − 1.40e6·15-s + 1.68e6·16-s − 9.92e6·17-s + 1.23e6·18-s − 1.60e7·19-s − 9.32e6·20-s − 5.38e6·21-s + 1.96e7·22-s − 9.19e6·23-s + 1.86e7·24-s − 1.51e7·25-s − 2.33e7·26-s − 1.43e7·27-s − 3.56e7·28-s + ⋯ |
| L(s) = 1 | + 0.463·2-s − 0.577·3-s − 0.785·4-s + 0.829·5-s − 0.267·6-s + 0.498·7-s − 0.827·8-s + 0.333·9-s + 0.384·10-s + 1.75·11-s + 0.453·12-s − 0.832·13-s + 0.231·14-s − 0.479·15-s + 0.401·16-s − 1.69·17-s + 0.154·18-s − 1.49·19-s − 0.651·20-s − 0.287·21-s + 0.812·22-s − 0.298·23-s + 0.477·24-s − 0.311·25-s − 0.385·26-s − 0.192·27-s − 0.391·28-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 | \( 1 - 20.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 5.79e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.21e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.36e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.11e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 9.92e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.60e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 9.19e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.93e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.28e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.39e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.59e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 5.30e6T + 9.29e17T^{2} \) |
| 47 | \( 1 - 9.16e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.20e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 8.56e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.23e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 4.22e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 6.29e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 5.22e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 7.62e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.11e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.48e11T + 7.15e21T^{2} \) |
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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
−10.06803755423998834034997516603, −9.253293045316136502814325475398, −8.362984368550223201619578924627, −6.54265276920972073378508737513, −6.11947937124597838906675477407, −4.53786911177669483302783797092, −4.38186852947030872884860017969, −2.46886523291150976120521769819, −1.25463636889220777619617458450, 0,
1.25463636889220777619617458450, 2.46886523291150976120521769819, 4.38186852947030872884860017969, 4.53786911177669483302783797092, 6.11947937124597838906675477407, 6.54265276920972073378508737513, 8.362984368550223201619578924627, 9.253293045316136502814325475398, 10.06803755423998834034997516603
