| L(s) = 1 | − 2-s + 6·3-s − 7·4-s + 16·5-s − 6·6-s − 2·7-s + 7·8-s + 27·9-s − 16·10-s − 42·12-s + 76·13-s + 2·14-s + 96·15-s − 7·16-s + 26·17-s − 27·18-s + 54·19-s − 112·20-s − 12·21-s + 224·23-s + 42·24-s + 74·25-s − 76·26-s + 108·27-s + 14·28-s − 222·29-s − 96·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1.15·3-s − 7/8·4-s + 1.43·5-s − 0.408·6-s − 0.107·7-s + 0.309·8-s + 9-s − 0.505·10-s − 1.01·12-s + 1.62·13-s + 0.0381·14-s + 1.65·15-s − 0.109·16-s + 0.370·17-s − 0.353·18-s + 0.652·19-s − 1.25·20-s − 0.124·21-s + 2.03·23-s + 0.357·24-s + 0.591·25-s − 0.573·26-s + 0.769·27-s + 0.0944·28-s − 1.42·29-s − 0.584·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.597162535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.597162535\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + p^{3} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 16 T + 182 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 26 T + 2570 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 54 T + 11774 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 222 T + 43642 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 48 T + 85910 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 494 T + 198818 T^{2} - 494 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 64 T + 189662 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 84 T + 164350 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 196 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 1104 T + 736358 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 928 T + 626214 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 456 T + 488494 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 592 T + 341742 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 230 T + 954126 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 348 T + 307798 T^{2} + 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 972 T + 1645606 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.95020522480700587635653227889, −10.88742624466023394016365768633, −9.898409227594244649761266404563, −9.691795110041662432003406481650, −9.240365588721621303282605442657, −9.225877669052750091781846659500, −8.506293879747501162777095628574, −8.207484895452689674703071830089, −7.67339622796756854528168352334, −6.87416537470346353747953660032, −6.61382802346307400930528768102, −5.84343263186674276172927335522, −5.30524229996646753735537767865, −4.96418730566925242607158430799, −3.79971373478751606103720714141, −3.78467198257850647314745477507, −2.84405679420415898961178958460, −2.22316395953988977196829648052, −1.38111615118597703042253123077, −0.836177341442496644460259406892,
0.836177341442496644460259406892, 1.38111615118597703042253123077, 2.22316395953988977196829648052, 2.84405679420415898961178958460, 3.78467198257850647314745477507, 3.79971373478751606103720714141, 4.96418730566925242607158430799, 5.30524229996646753735537767865, 5.84343263186674276172927335522, 6.61382802346307400930528768102, 6.87416537470346353747953660032, 7.67339622796756854528168352334, 8.207484895452689674703071830089, 8.506293879747501162777095628574, 9.225877669052750091781846659500, 9.240365588721621303282605442657, 9.691795110041662432003406481650, 9.898409227594244649761266404563, 10.88742624466023394016365768633, 10.95020522480700587635653227889
