| L(s) = 1 | + 33.0·2-s + 243·3-s − 957.·4-s − 9.84e3·5-s + 8.02e3·6-s + 6.60e4·7-s − 9.92e4·8-s + 5.90e4·9-s − 3.24e5·10-s − 9.49e5·11-s − 2.32e5·12-s + 2.41e6·13-s + 2.17e6·14-s − 2.39e6·15-s − 1.31e6·16-s + 3.48e6·17-s + 1.94e6·18-s + 4.12e6·19-s + 9.42e6·20-s + 1.60e7·21-s − 3.13e7·22-s + 6.06e6·23-s − 2.41e7·24-s + 4.80e7·25-s + 7.96e7·26-s + 1.43e7·27-s − 6.32e7·28-s + ⋯ |
| L(s) = 1 | + 0.729·2-s + 0.577·3-s − 0.467·4-s − 1.40·5-s + 0.421·6-s + 1.48·7-s − 1.07·8-s + 0.333·9-s − 1.02·10-s − 1.77·11-s − 0.269·12-s + 1.80·13-s + 1.08·14-s − 0.813·15-s − 0.313·16-s + 0.595·17-s + 0.243·18-s + 0.381·19-s + 0.658·20-s + 0.857·21-s − 1.29·22-s + 0.196·23-s − 0.618·24-s + 0.983·25-s + 1.31·26-s + 0.192·27-s − 0.694·28-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 | \( 1 - 33.0T + 2.04e3T^{2} \) |
| 5 | \( 1 + 9.84e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 6.60e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.49e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.41e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 3.48e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.12e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 6.06e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 2.69e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.69e6T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.82e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 8.77e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.13e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.51e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.99e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 2.72e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.65e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.39e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 2.31e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.76e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.98e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.08e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.23e10T + 7.15e21T^{2} \) |
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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
−10.41531889151699702052023906635, −8.597313773493366494384975089406, −8.286966823840981358980624796332, −7.43559107334063965519567511073, −5.56557887638284668778137360564, −4.73217220293984252123444473662, −3.80959696687344374970682254840, −2.98352421610975236494133123191, −1.30713781877628312156761689274, 0,
1.30713781877628312156761689274, 2.98352421610975236494133123191, 3.80959696687344374970682254840, 4.73217220293984252123444473662, 5.56557887638284668778137360564, 7.43559107334063965519567511073, 8.286966823840981358980624796332, 8.597313773493366494384975089406, 10.41531889151699702052023906635
