| L(s) = 1 | − 54·2-s − 120·3-s + 1.46e3·4-s + 1.35e4·5-s + 6.48e3·6-s − 1.10e5·8-s − 2.22e5·9-s − 7.29e5·10-s − 7.50e5·11-s − 1.75e5·12-s + 9.54e3·13-s − 1.62e6·15-s + 5.63e6·16-s − 4.16e6·17-s + 1.19e7·18-s + 1.79e7·19-s + 1.97e7·20-s + 4.05e7·22-s − 6.61e7·23-s + 1.32e7·24-s + 3.93e7·25-s − 5.15e5·26-s + 3.38e7·27-s + 6.15e7·29-s + 8.74e7·30-s + 1.52e7·31-s − 7.79e7·32-s + ⋯ |
| L(s) = 1 | − 1.19·2-s − 0.285·3-s + 0.712·4-s + 1.93·5-s + 0.340·6-s − 1.19·8-s − 1.25·9-s − 2.30·10-s − 1.40·11-s − 0.203·12-s + 0.00713·13-s − 0.550·15-s + 1.34·16-s − 0.710·17-s + 1.49·18-s + 1.66·19-s + 1.37·20-s + 1.67·22-s − 2.14·23-s + 0.340·24-s + 0.806·25-s − 0.00851·26-s + 0.453·27-s + 0.556·29-s + 0.657·30-s + 0.0958·31-s − 0.410·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+11/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + 27 p T + 91 p^{4} T^{2} + 27 p^{12} T^{3} + p^{22} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 40 p T + 26290 p^{2} T^{2} + 40 p^{12} T^{3} + p^{22} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 108 p^{3} T + 5715274 p^{2} T^{2} - 108 p^{14} T^{3} + p^{22} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 68256 p T + 653251941286 T^{2} + 68256 p^{12} T^{3} + p^{22} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9548 T + 580074739914 T^{2} - 9548 p^{11} T^{3} + p^{22} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4160052 T + 72837924685942 T^{2} + 4160052 p^{11} T^{3} + p^{22} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 17998712 T + 293399338219938 T^{2} - 17998712 p^{11} T^{3} + p^{22} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 66161016 T + 2994683043115918 T^{2} + 66161016 p^{11} T^{3} + p^{22} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2121228 p T + 12823193731307518 T^{2} - 2121228 p^{12} T^{3} + p^{22} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 15281552 T + 44544265736191854 T^{2} - 15281552 p^{11} T^{3} + p^{22} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 527218340 T + 315935809764623790 T^{2} + 527218340 p^{11} T^{3} + p^{22} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 178276140 T + 1103495454485680198 T^{2} - 178276140 p^{11} T^{3} + p^{22} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1826745232 T + 2408495597390567334 T^{2} - 1826745232 p^{11} T^{3} + p^{22} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 568240704 T + 125222806434661774 T^{2} + 568240704 p^{11} T^{3} + p^{22} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4185816372 T + 8708326678160294206 T^{2} + 4185816372 p^{11} T^{3} + p^{22} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3111345000 T + 25961868574953911218 T^{2} + 3111345000 p^{11} T^{3} + p^{22} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15042595060 T + \)\(13\!\cdots\!78\)\( T^{2} + 15042595060 p^{11} T^{3} + p^{22} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9856523968 T + \)\(12\!\cdots\!18\)\( T^{2} - 9856523968 p^{11} T^{3} + p^{22} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 24312011328 T + \)\(59\!\cdots\!02\)\( T^{2} + 24312011328 p^{11} T^{3} + p^{22} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 30890001932 T + \)\(73\!\cdots\!46\)\( T^{2} - 30890001932 p^{11} T^{3} + p^{22} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1992804256 T + \)\(85\!\cdots\!38\)\( T^{2} - 1992804256 p^{11} T^{3} + p^{22} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5277014568 T + \)\(22\!\cdots\!46\)\( T^{2} + 5277014568 p^{11} T^{3} + p^{22} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 101541312828 T + \)\(76\!\cdots\!78\)\( T^{2} - 101541312828 p^{11} T^{3} + p^{22} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 192228621116 T + \)\(23\!\cdots\!14\)\( T^{2} - 192228621116 p^{11} T^{3} + p^{22} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.95776021716272696957431266282, −12.07565486470200023105733234463, −11.79616546142252160558415392976, −10.86603076261353902873417974893, −10.35274861449342546499229175346, −9.902057681664807588427017111538, −9.181392927418900040200293843283, −9.118239769779742969999386994405, −7.976282909673634197872119276196, −7.80301022621852801861021199699, −6.43269232891223512667039671578, −5.95677381188123483946378908192, −5.66436002545152060946769471324, −4.93394815933968296097972652090, −3.33487864033064992997976933759, −2.53294342489107854684642164936, −2.16720861015607171635815597964, −1.21140542763281010288777580288, 0, 0,
1.21140542763281010288777580288, 2.16720861015607171635815597964, 2.53294342489107854684642164936, 3.33487864033064992997976933759, 4.93394815933968296097972652090, 5.66436002545152060946769471324, 5.95677381188123483946378908192, 6.43269232891223512667039671578, 7.80301022621852801861021199699, 7.976282909673634197872119276196, 9.118239769779742969999386994405, 9.181392927418900040200293843283, 9.902057681664807588427017111538, 10.35274861449342546499229175346, 10.86603076261353902873417974893, 11.79616546142252160558415392976, 12.07565486470200023105733234463, 12.95776021716272696957431266282
