| L(s) = 1 | + 4·2-s + 16·4-s + 14·5-s − 170·7-s + 64·8-s + 56·10-s + 250·11-s − 169·13-s − 680·14-s + 256·16-s − 1.06e3·17-s − 78·19-s + 224·20-s + 1.00e3·22-s − 1.57e3·23-s − 2.92e3·25-s − 676·26-s − 2.72e3·28-s − 2.57e3·29-s − 8.65e3·31-s + 1.02e3·32-s − 4.24e3·34-s − 2.38e3·35-s + 1.09e4·37-s − 312·38-s + 896·40-s − 1.05e3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.250·5-s − 1.31·7-s + 0.353·8-s + 0.177·10-s + 0.622·11-s − 0.277·13-s − 0.927·14-s + 1/4·16-s − 0.891·17-s − 0.0495·19-s + 0.125·20-s + 0.440·22-s − 0.621·23-s − 0.937·25-s − 0.196·26-s − 0.655·28-s − 0.569·29-s − 1.61·31-s + 0.176·32-s − 0.630·34-s − 0.328·35-s + 1.31·37-s − 0.0350·38-s + 0.0885·40-s − 0.0975·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 + 170 T + p^{5} T^{2} \) |
| 11 | \( 1 - 250 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1062 T + p^{5} T^{2} \) |
| 19 | \( 1 + 78 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1576 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2578 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8654 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10986 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1050 T + p^{5} T^{2} \) |
| 43 | \( 1 + 5900 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5962 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29046 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13922 T + p^{5} T^{2} \) |
| 61 | \( 1 + 32882 T + p^{5} T^{2} \) |
| 67 | \( 1 + 69566 T + p^{5} T^{2} \) |
| 71 | \( 1 - 50542 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46750 T + p^{5} T^{2} \) |
| 79 | \( 1 + 19348 T + p^{5} T^{2} \) |
| 83 | \( 1 - 87438 T + p^{5} T^{2} \) |
| 89 | \( 1 + 94170 T + p^{5} T^{2} \) |
| 97 | \( 1 - 182786 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96286446143680722125998040716, −9.811940174037741267880492204866, −9.110867397245520150626416888061, −7.55006747926477459292837023913, −6.50064778054188545329943562289, −5.80456699072340915757753756135, −4.31479327546449218348073586296, −3.29815951405891859265258186166, −1.96651068802625974033326410568, 0,
1.96651068802625974033326410568, 3.29815951405891859265258186166, 4.31479327546449218348073586296, 5.80456699072340915757753756135, 6.50064778054188545329943562289, 7.55006747926477459292837023913, 9.110867397245520150626416888061, 9.811940174037741267880492204866, 10.96286446143680722125998040716
