| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 10·5-s + 24·6-s − 30·7-s + 32·8-s + 27·9-s + 40·10-s − 77·11-s + 72·12-s − 36·13-s − 120·14-s + 60·15-s + 80·16-s − 65·17-s + 108·18-s − 55·19-s + 120·20-s − 180·21-s − 308·22-s − 128·23-s + 192·24-s + 75·25-s − 144·26-s + 108·27-s − 360·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.61·7-s + 1.41·8-s + 9-s + 1.26·10-s − 2.11·11-s + 1.73·12-s − 0.768·13-s − 2.29·14-s + 1.03·15-s + 5/4·16-s − 0.927·17-s + 1.41·18-s − 0.664·19-s + 1.34·20-s − 1.87·21-s − 2.98·22-s − 1.16·23-s + 1.63·24-s + 3/5·25-s − 1.08·26-s + 0.769·27-s − 2.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 37 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 15 T + p^{3} T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 7 p T + 4007 T^{2} + 7 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 36 T + 3193 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 65 T + 10501 T^{2} + 65 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 55 T + 8969 T^{2} + 55 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 128 T + 18121 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 53 T + 12865 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 388 T + 86909 T^{2} + 388 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 118 T + 12247 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 353 T + 147329 T^{2} + 353 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 146 T + 209071 T^{2} + 146 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 448 T + 128330 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 17 T + 34933 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 383527 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 147 T + 525661 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 45 T + 584431 T^{2} + 45 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 163 T + 750989 T^{2} - 163 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 53 T + 975663 T^{2} + 53 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1389 T + 1372129 T^{2} + 1389 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1018 T + 1334983 T^{2} + 1018 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2318 T + 3080543 T^{2} - 2318 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.208502415486305368825616976892, −9.067390319827868482280830045114, −8.215241971356809728310252408823, −8.082150181684514821760595250866, −7.31401712948829052649502822530, −7.27534755602640002920595316717, −6.49911929555020335862029939160, −6.43016238123071192543092571516, −5.64883855075278718172805936018, −5.54158566327720649219651126236, −4.73061997157057649994681705515, −4.65056697063125303034925956830, −3.62312777389315297434712970495, −3.57773536804614854620816769602, −2.74163251680221382164206737512, −2.71833802875145940837054996106, −1.88738428039713629892708528973, −1.88138243943507535629368002119, 0, 0,
1.88138243943507535629368002119, 1.88738428039713629892708528973, 2.71833802875145940837054996106, 2.74163251680221382164206737512, 3.57773536804614854620816769602, 3.62312777389315297434712970495, 4.65056697063125303034925956830, 4.73061997157057649994681705515, 5.54158566327720649219651126236, 5.64883855075278718172805936018, 6.43016238123071192543092571516, 6.49911929555020335862029939160, 7.27534755602640002920595316717, 7.31401712948829052649502822530, 8.082150181684514821760595250866, 8.215241971356809728310252408823, 9.067390319827868482280830045114, 9.208502415486305368825616976892
