| L(s) = 1 | + 81.8·2-s + 243·3-s + 4.64e3·4-s − 1.04e4·5-s + 1.98e4·6-s − 3.31e4·7-s + 2.12e5·8-s + 5.90e4·9-s − 8.54e5·10-s + 6.92e5·11-s + 1.12e6·12-s − 1.94e6·13-s − 2.71e6·14-s − 2.53e6·15-s + 7.90e6·16-s + 9.86e6·17-s + 4.83e6·18-s − 8.54e6·19-s − 4.85e7·20-s − 8.06e6·21-s + 5.67e7·22-s − 1.31e7·23-s + 5.17e7·24-s + 6.00e7·25-s − 1.58e8·26-s + 1.43e7·27-s − 1.54e8·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 0.577·3-s + 2.27·4-s − 1.49·5-s + 1.04·6-s − 0.746·7-s + 2.29·8-s + 0.333·9-s − 2.70·10-s + 1.29·11-s + 1.31·12-s − 1.44·13-s − 1.34·14-s − 0.862·15-s + 1.88·16-s + 1.68·17-s + 0.602·18-s − 0.791·19-s − 3.39·20-s − 0.430·21-s + 2.34·22-s − 0.424·23-s + 1.32·24-s + 1.23·25-s − 2.62·26-s + 0.192·27-s − 1.69·28-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 | \( 1 - 81.8T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.04e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.31e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 6.92e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.94e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 9.86e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.54e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.31e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.43e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.66e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.10e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 8.43e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.21e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 9.46e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.89e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 9.26e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.35e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.44e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.73e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 8.03e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.54e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.14e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.11e11T + 7.15e21T^{2} \) |
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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
−10.49555590835467881132248739125, −9.135262607741950285900713058170, −7.53300390868595038973303151047, −7.12938016931803311932764946308, −5.80639215924175946561683908905, −4.53003676378456576405259443099, −3.66212336393471813353920385962, −3.25661953692079311206709582357, −1.82784342811883613989702407655, 0,
1.82784342811883613989702407655, 3.25661953692079311206709582357, 3.66212336393471813353920385962, 4.53003676378456576405259443099, 5.80639215924175946561683908905, 7.12938016931803311932764946308, 7.53300390868595038973303151047, 9.135262607741950285900713058170, 10.49555590835467881132248739125
