| L(s) = 1 | + 4·2-s − 6·3-s + 12·4-s + 10·5-s − 24·6-s + 32·8-s + 27·9-s + 40·10-s − 20·11-s − 72·12-s + 42·13-s − 60·15-s + 80·16-s + 76·17-s + 108·18-s + 90·19-s + 120·20-s − 80·22-s + 44·23-s − 192·24-s + 75·25-s + 168·26-s − 108·27-s − 160·29-s − 240·30-s + 62·31-s + 192·32-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s + 1.26·10-s − 0.548·11-s − 1.73·12-s + 0.896·13-s − 1.03·15-s + 5/4·16-s + 1.08·17-s + 1.41·18-s + 1.08·19-s + 1.34·20-s − 0.775·22-s + 0.398·23-s − 1.63·24-s + 3/5·25-s + 1.26·26-s − 0.769·27-s − 1.02·29-s − 1.46·30-s + 0.359·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.70659548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.70659548\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 20 T + 1612 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 42 T + 4789 T^{2} - 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 76 T + 5704 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 90 T + 15559 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 19252 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 160 T + 54028 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 p T + 57599 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 358 T + 89141 T^{2} + 358 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 36 T + 2552 p T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 134 T + 129969 T^{2} + 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 684 T + 264994 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 16 T + 182818 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 552 T + 476584 T^{2} - 552 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1312 T + 869394 T^{2} - 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 194 T + 533609 T^{2} - 194 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 380 T - 111452 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 198 T + 782269 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 126 T - 277529 T^{2} - 126 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 864 T + 336184 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 184 T + 284548 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 568 T + 1864602 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.270637643037467260850064526427, −9.264051974263001704443566347563, −8.240950244635466580259852305238, −8.201786988841597633647076574782, −7.29558339210911692524442609049, −7.27606047285164156058641187016, −6.58624307924446828377669629216, −6.51844260159011980944332036582, −5.68490403038104167537801254806, −5.57435402161907771468462835892, −5.21606602154454300269117767983, −5.20802082199638658860788932152, −4.19233550056785946849025900572, −4.00040335056452688784727922145, −3.29599912069091022356339471186, −3.04915767295846506955227300356, −2.10182669192227319031896953335, −1.88868067312090816918325056299, −0.926673912049234269661883515813, −0.77002131172033956884370680772,
0.77002131172033956884370680772, 0.926673912049234269661883515813, 1.88868067312090816918325056299, 2.10182669192227319031896953335, 3.04915767295846506955227300356, 3.29599912069091022356339471186, 4.00040335056452688784727922145, 4.19233550056785946849025900572, 5.20802082199638658860788932152, 5.21606602154454300269117767983, 5.57435402161907771468462835892, 5.68490403038104167537801254806, 6.51844260159011980944332036582, 6.58624307924446828377669629216, 7.27606047285164156058641187016, 7.29558339210911692524442609049, 8.201786988841597633647076574782, 8.240950244635466580259852305238, 9.264051974263001704443566347563, 9.270637643037467260850064526427
