| L(s) = 1 | + 51.1·2-s − 728.·3-s + 571.·4-s − 4.17e3·5-s − 3.73e4·6-s − 8.85e3·7-s − 7.55e4·8-s + 3.54e5·9-s − 2.13e5·10-s + 8.78e4·11-s − 4.16e5·12-s − 5.59e5·13-s − 4.53e5·14-s + 3.04e6·15-s − 5.03e6·16-s + 9.36e6·17-s + 1.81e7·18-s + 1.25e7·19-s − 2.38e6·20-s + 6.45e6·21-s + 4.49e6·22-s + 2.73e7·23-s + 5.50e7·24-s − 3.14e7·25-s − 2.86e7·26-s − 1.28e8·27-s − 5.05e6·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 1.73·3-s + 0.278·4-s − 0.597·5-s − 1.95·6-s − 0.199·7-s − 0.815·8-s + 1.99·9-s − 0.675·10-s + 0.164·11-s − 0.483·12-s − 0.417·13-s − 0.225·14-s + 1.03·15-s − 1.20·16-s + 1.59·17-s + 2.26·18-s + 1.16·19-s − 0.166·20-s + 0.344·21-s + 0.185·22-s + 0.886·23-s + 1.41·24-s − 0.643·25-s − 0.472·26-s − 1.72·27-s − 0.0555·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.259046479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259046479\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 - 51.1T + 2.04e3T^{2} \) |
| 3 | \( 1 + 728.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 4.17e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 8.85e3T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.78e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 5.59e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 9.36e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.25e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.73e7T + 9.52e14T^{2} \) |
| 31 | \( 1 + 1.62e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.12e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 5.54e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.97e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.66e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.50e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.33e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.05e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 4.18e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 9.18e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.04e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.36e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.15e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 5.67e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.68e10T + 7.15e21T^{2} \) |
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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
−14.54006345602472620549279639374, −12.97111235025176163406824760769, −12.06639370602696829518772706886, −11.38804131017301409740534184412, −9.774148808147995530456580917634, −7.28613138152425670570617570382, −5.82776827847138552074698671509, −5.00230611418909599821835806079, −3.59844552284227190646038744183, −0.69743778295074663274327102426,
0.69743778295074663274327102426, 3.59844552284227190646038744183, 5.00230611418909599821835806079, 5.82776827847138552074698671509, 7.28613138152425670570617570382, 9.774148808147995530456580917634, 11.38804131017301409740534184412, 12.06639370602696829518772706886, 12.97111235025176163406824760769, 14.54006345602472620549279639374
