| L(s) = 1 | + 5·2-s − 9·4-s + 50·5-s + 80·7-s − 55·8-s + 250·10-s + 800·11-s − 120·13-s + 400·14-s − 193·16-s + 1.94e3·17-s − 1.51e3·19-s − 450·20-s + 4.00e3·22-s − 1.32e3·23-s + 1.87e3·25-s − 600·26-s − 720·28-s + 1.30e3·29-s − 5.82e3·31-s − 5.65e3·32-s + 9.70e3·34-s + 4.00e3·35-s − 1.25e4·37-s − 7.56e3·38-s − 2.75e3·40-s + 400·41-s + ⋯ |
| L(s) = 1 | + 0.883·2-s − 0.281·4-s + 0.894·5-s + 0.617·7-s − 0.303·8-s + 0.790·10-s + 1.99·11-s − 0.196·13-s + 0.545·14-s − 0.188·16-s + 1.62·17-s − 0.960·19-s − 0.251·20-s + 1.76·22-s − 0.520·23-s + 3/5·25-s − 0.174·26-s − 0.173·28-s + 0.287·29-s − 1.08·31-s − 0.976·32-s + 1.43·34-s + 0.551·35-s − 1.50·37-s − 0.849·38-s − 0.271·40-s + 0.0371·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.313009369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.313009369\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - 5 T + 17 p T^{2} - 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 80 T + 20714 T^{2} - 80 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 800 T + 467602 T^{2} - 800 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 120 T + 514186 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1940 T + 3743494 T^{2} - 1940 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1512 T + 1811734 T^{2} + 1512 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1320 T + 4100206 T^{2} + 1320 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 1300 T - 17947202 T^{2} - 1300 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5824 T + 58720046 T^{2} + 5824 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12560 T + 102958314 T^{2} + 12560 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 400 T + 164704402 T^{2} - 400 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 25680 T + 399490486 T^{2} - 25680 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18920 T + 546954334 T^{2} - 18920 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 49460 T + 1434563566 T^{2} - 49460 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 63200 T + 2393594098 T^{2} - 63200 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 49116 T + 1219519966 T^{2} + 49116 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6080 T + 319369814 T^{2} - 6080 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 65200 T + 4591816702 T^{2} - 65200 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 97740 T + 5753972086 T^{2} - 97740 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 46288 T + 6429395534 T^{2} + 46288 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 57360 T + 4528611766 T^{2} + 57360 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 87000 T + 10502568898 T^{2} + 87000 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10180 T + 16899916614 T^{2} - 10180 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−14.66784970969022019510655850936, −14.37784536888453374327419702333, −14.03683634096722307094210643198, −13.68496200415498815155107445593, −12.61180369256250028057802710003, −12.48918967488013551504117405756, −11.77617916094901859396700299413, −11.01034771856315258632575862221, −10.32377565394036678169206692921, −9.596994509059774957244181837257, −9.023182964180726345828141504177, −8.460563802580778649482788226914, −7.35453074034000674602867122277, −6.66667448731695673263077309964, −5.70691639151721770956250193902, −5.29318868746665421033732889688, −4.15975404841838252730105105759, −3.76406847311085762487158546967, −2.13598064014564555158892296757, −1.11486975673290849705631102973,
1.11486975673290849705631102973, 2.13598064014564555158892296757, 3.76406847311085762487158546967, 4.15975404841838252730105105759, 5.29318868746665421033732889688, 5.70691639151721770956250193902, 6.66667448731695673263077309964, 7.35453074034000674602867122277, 8.460563802580778649482788226914, 9.023182964180726345828141504177, 9.596994509059774957244181837257, 10.32377565394036678169206692921, 11.01034771856315258632575862221, 11.77617916094901859396700299413, 12.48918967488013551504117405756, 12.61180369256250028057802710003, 13.68496200415498815155107445593, 14.03683634096722307094210643198, 14.37784536888453374327419702333, 14.66784970969022019510655850936
