| L(s) = 1 | + 2·5-s + 7-s − 3·9-s + 11-s + 13-s − 2·17-s − 4·19-s − 25-s − 6·29-s + 4·31-s + 2·35-s − 6·37-s + 6·41-s + 12·43-s − 6·45-s − 4·47-s + 49-s + 10·53-s + 2·55-s − 10·61-s − 3·63-s + 2·65-s + 12·67-s + 14·73-s + 77-s + 9·81-s − 4·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s + 0.301·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.986·37-s + 0.937·41-s + 1.82·43-s − 0.894·45-s − 0.583·47-s + 1/7·49-s + 1.37·53-s + 0.269·55-s − 1.28·61-s − 0.377·63-s + 0.248·65-s + 1.46·67-s + 1.63·73-s + 0.113·77-s + 81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.35373565220657, −13.89645266284317, −13.72503608697611, −13.00443436469241, −12.49882019300086, −12.02138484448040, −11.27350316167251, −10.99472712603128, −10.58534292229963, −9.788202368638226, −9.371373265589957, −8.884905440599308, −8.369225789295129, −7.918880950319420, −7.121760813324626, −6.586399850005408, −5.972879424419328, −5.645004317236390, −5.095983901692654, −4.234508953747260, −3.860672628508289, −2.919957440088450, −2.318899358778175, −1.870136885052319, −0.9776934884153040, 0,
0.9776934884153040, 1.870136885052319, 2.318899358778175, 2.919957440088450, 3.860672628508289, 4.234508953747260, 5.095983901692654, 5.645004317236390, 5.972879424419328, 6.586399850005408, 7.121760813324626, 7.918880950319420, 8.369225789295129, 8.884905440599308, 9.371373265589957, 9.788202368638226, 10.58534292229963, 10.99472712603128, 11.27350316167251, 12.02138484448040, 12.49882019300086, 13.00443436469241, 13.72503608697611, 13.89645266284317, 14.35373565220657
