| L(s) = 1 | + 4·2-s − 3-s + 12·4-s − 4·6-s − 57·7-s + 32·8-s − 9·9-s + 10·11-s − 12·12-s − 13·13-s − 228·14-s + 80·16-s + 51·17-s − 36·18-s − 38·19-s + 57·21-s + 40·22-s + 155·23-s − 32·24-s − 52·26-s − 8·27-s − 684·28-s − 79·29-s − 16·31-s + 192·32-s − 10·33-s + 204·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.272·6-s − 3.07·7-s + 1.41·8-s − 1/3·9-s + 0.274·11-s − 0.288·12-s − 0.277·13-s − 4.35·14-s + 5/4·16-s + 0.727·17-s − 0.471·18-s − 0.458·19-s + 0.592·21-s + 0.387·22-s + 1.40·23-s − 0.272·24-s − 0.392·26-s − 0.0570·27-s − 4.61·28-s − 0.505·29-s − 0.0926·31-s + 1.06·32-s − 0.0527·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 902500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 10 T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 57 T + 1454 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + p T + 2268 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 p T + 520 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 155 T + 994 p T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 79 T + 13124 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 16 T + 48318 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 380 T + 126078 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 296 T + 78966 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 200 T + 146846 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 397 T + 333572 T^{2} + 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 680 T + 483894 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 939 T + 740138 T^{2} - 939 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 406 T + 735614 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 123 T + 781772 T^{2} + 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 106 T + 840030 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2226 T + 2380750 T^{2} + 2226 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1864 T + 2438382 T^{2} - 1864 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.629576398992878818528042059212, −9.194662380355910187118361404389, −8.525297720716282184656598593212, −8.342474162730769681284093307756, −7.30597435772406113342176953640, −6.93934498463053888675641504050, −6.88083544045976845019605304651, −6.46577391308989674088085143348, −5.84585713140762792056314855940, −5.69854559396613702682872783528, −5.07269518462082386258986104834, −4.66408454669520975301503332128, −3.72840049233367941783150918293, −3.56001818230604933785408208542, −3.14949896343821601494171024124, −2.87917478794303451032666866285, −2.05745957732913278040109158479, −1.24035554946339237114310749178, 0, 0,
1.24035554946339237114310749178, 2.05745957732913278040109158479, 2.87917478794303451032666866285, 3.14949896343821601494171024124, 3.56001818230604933785408208542, 3.72840049233367941783150918293, 4.66408454669520975301503332128, 5.07269518462082386258986104834, 5.69854559396613702682872783528, 5.84585713140762792056314855940, 6.46577391308989674088085143348, 6.88083544045976845019605304651, 6.93934498463053888675641504050, 7.30597435772406113342176953640, 8.342474162730769681284093307756, 8.525297720716282184656598593212, 9.194662380355910187118361404389, 9.629576398992878818528042059212
