| L(s) = 1 | + 8·2-s − 22·3-s + 48·4-s + 68·5-s − 176·6-s + 312·7-s + 256·8-s − 51·9-s + 544·10-s + 12·11-s − 1.05e3·12-s − 1.48e3·13-s + 2.49e3·14-s − 1.49e3·15-s + 1.28e3·16-s − 1.76e3·17-s − 408·18-s + 1.12e3·19-s + 3.26e3·20-s − 6.86e3·21-s + 96·22-s − 5.63e3·24-s + 2.21e3·25-s − 1.18e4·26-s + 7.54e3·27-s + 1.49e4·28-s − 5.11e3·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.41·3-s + 3/2·4-s + 1.21·5-s − 1.99·6-s + 2.40·7-s + 1.41·8-s − 0.209·9-s + 1.72·10-s + 0.0299·11-s − 2.11·12-s − 2.43·13-s + 3.40·14-s − 1.71·15-s + 5/4·16-s − 1.48·17-s − 0.296·18-s + 0.714·19-s + 1.82·20-s − 3.39·21-s + 0.0422·22-s − 1.99·24-s + 0.709·25-s − 3.43·26-s + 1.99·27-s + 3.60·28-s − 1.13·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 22 T + 535 T^{2} + 22 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 68 T + 2406 T^{2} - 68 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 312 T + 55638 T^{2} - 312 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 12 T - 165934 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 114 p T + 1291155 T^{2} + 114 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 104 p T + 2701802 T^{2} + 104 p^{6} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1124 T + 4935114 T^{2} - 1124 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5118 T + 17025851 T^{2} + 5118 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3110 T + 55188319 T^{2} + 3110 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 36 T - 28025562 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7234 T + 240366803 T^{2} + 7234 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 28816 T + 478269238 T^{2} + 28816 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 19918 T + 555189767 T^{2} + 19918 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8508 T + 329866670 T^{2} - 8508 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 46944 T + 1978147574 T^{2} + 46944 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 61860 T + 2240302302 T^{2} - 61860 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 25428 T + 2388191810 T^{2} - 25428 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 49974 T + 940309919 T^{2} + 49974 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 11610 T + 3753754779 T^{2} - 11610 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 11340 T + 5210502210 T^{2} - 11340 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16352 T + 5457546750 T^{2} - 16352 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 126600 T + 14382988898 T^{2} - 126600 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 67744 T + 18289446586 T^{2} + 67744 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−8.797906159682830529792737468759, −8.631249478825411970777402616959, −7.81059691996544883946720383882, −7.72917229502641559929227636429, −6.94732245194646183345204999651, −6.80489954829061762226670457585, −6.22424532417916487257879492922, −5.78200887162413002571564606743, −5.23313958249785947009111794185, −5.01906020989443615996363847039, −4.99655421134761404768986208190, −4.65954468223570258341400498005, −3.85229506564377259769667481524, −3.10755913816638856128407680362, −2.35606685300579270153562150451, −2.20818804611360809661663714286, −1.68020225102863261484957530157, −1.22000232525028198634011706567, 0, 0,
1.22000232525028198634011706567, 1.68020225102863261484957530157, 2.20818804611360809661663714286, 2.35606685300579270153562150451, 3.10755913816638856128407680362, 3.85229506564377259769667481524, 4.65954468223570258341400498005, 4.99655421134761404768986208190, 5.01906020989443615996363847039, 5.23313958249785947009111794185, 5.78200887162413002571564606743, 6.22424532417916487257879492922, 6.80489954829061762226670457585, 6.94732245194646183345204999651, 7.72917229502641559929227636429, 7.81059691996544883946720383882, 8.631249478825411970777402616959, 8.797906159682830529792737468759
