| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s − 4·10-s − 2·11-s + 6·12-s − 4·14-s + 4·15-s + 5·16-s − 4·17-s − 6·18-s − 12·19-s + 6·20-s + 4·21-s + 4·22-s + 2·23-s − 8·24-s + 3·25-s + 4·27-s + 6·28-s + 4·29-s − 8·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s − 0.603·11-s + 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s − 0.970·17-s − 1.41·18-s − 2.75·19-s + 1.34·20-s + 0.872·21-s + 0.852·22-s + 0.417·23-s − 1.63·24-s + 3/5·25-s + 0.769·27-s + 1.13·28-s + 0.742·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.974380154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.974380154\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 90 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 46 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 102 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 162 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.324879362536268305226009075557, −8.295858778831265648647717397307, −7.76190986503516542735774339304, −7.68465894745741975349169286599, −6.96850114688887152038128490205, −6.71162115505676129564793190841, −6.43183015426782933376632376441, −6.22276615654099260342162773954, −5.40504721912048710630751567545, −5.32913397120662782787975404364, −4.54507390340573729070352715934, −4.41941654093257286246307742227, −3.84071302215742548291349741414, −3.38371555215605813375835115957, −2.67260669562618553857035774741, −2.30076780547346667120114416079, −2.10474281555997647732527398179, −1.98041498385356183753731884054, −1.02025477056700170093807526007, −0.59018947487253063624918640148,
0.59018947487253063624918640148, 1.02025477056700170093807526007, 1.98041498385356183753731884054, 2.10474281555997647732527398179, 2.30076780547346667120114416079, 2.67260669562618553857035774741, 3.38371555215605813375835115957, 3.84071302215742548291349741414, 4.41941654093257286246307742227, 4.54507390340573729070352715934, 5.32913397120662782787975404364, 5.40504721912048710630751567545, 6.22276615654099260342162773954, 6.43183015426782933376632376441, 6.71162115505676129564793190841, 6.96850114688887152038128490205, 7.68465894745741975349169286599, 7.76190986503516542735774339304, 8.295858778831265648647717397307, 8.324879362536268305226009075557
