| L(s) = 1 | − 270·5-s − 1.11e3·7-s − 5.72e3·11-s − 4.57e3·13-s + 3.65e4·17-s − 5.17e4·19-s + 2.22e4·23-s − 5.22e3·25-s + 1.57e5·29-s + 1.03e5·31-s + 3.00e5·35-s − 9.48e4·37-s − 6.59e5·41-s + 7.57e4·43-s + 4.05e5·47-s + 4.13e5·49-s + 1.34e6·53-s + 1.54e6·55-s − 1.30e6·59-s + 1.83e6·61-s + 1.23e6·65-s − 1.36e6·67-s + 2.71e6·71-s + 2.86e6·73-s + 6.36e6·77-s + 1.12e6·79-s + 5.91e6·83-s + ⋯ |
| L(s) = 1 | − 0.965·5-s − 1.22·7-s − 1.29·11-s − 0.576·13-s + 1.80·17-s − 1.73·19-s + 0.381·23-s − 0.0668·25-s + 1.19·29-s + 0.626·31-s + 1.18·35-s − 0.307·37-s − 1.49·41-s + 0.145·43-s + 0.569·47-s + 0.501·49-s + 1.24·53-s + 1.25·55-s − 0.826·59-s + 1.03·61-s + 0.557·65-s − 0.556·67-s + 0.899·71-s + 0.863·73-s + 1.58·77-s + 0.257·79-s + 1.13·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.7450879553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7450879553\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 54 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1112 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5724 T + p^{7} T^{2} \) |
| 13 | \( 1 + 4570 T + p^{7} T^{2} \) |
| 17 | \( 1 - 36558 T + p^{7} T^{2} \) |
| 19 | \( 1 + 51740 T + p^{7} T^{2} \) |
| 23 | \( 1 - 22248 T + p^{7} T^{2} \) |
| 29 | \( 1 - 157194 T + p^{7} T^{2} \) |
| 31 | \( 1 - 103936 T + p^{7} T^{2} \) |
| 37 | \( 1 + 94834 T + p^{7} T^{2} \) |
| 41 | \( 1 + 659610 T + p^{7} T^{2} \) |
| 43 | \( 1 - 75772 T + p^{7} T^{2} \) |
| 47 | \( 1 - 405648 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1346274 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1303884 T + p^{7} T^{2} \) |
| 61 | \( 1 - 30062 p T + p^{7} T^{2} \) |
| 67 | \( 1 + 1369388 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2714040 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2868794 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1129648 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5912028 T + p^{7} T^{2} \) |
| 89 | \( 1 - 897750 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13719074 T + p^{7} T^{2} \) |
| 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
−12.03944040680793216726893840988, −10.54359844775308336758883427917, −9.943653540384377823857189933523, −8.459850144321774823017567888803, −7.60049363859624918181539666513, −6.45526513890212086058298995863, −5.09429078990005493400703346456, −3.69013738432820489724785638457, −2.64905734908050382120198400344, −0.46876497425959232852878806690,
0.46876497425959232852878806690, 2.64905734908050382120198400344, 3.69013738432820489724785638457, 5.09429078990005493400703346456, 6.45526513890212086058298995863, 7.60049363859624918181539666513, 8.459850144321774823017567888803, 9.943653540384377823857189933523, 10.54359844775308336758883427917, 12.03944040680793216726893840988
