| L(s) = 1 | + 18·3-s + 160·5-s + 343·7-s − 1.86e3·9-s − 5.70e3·11-s + 1.38e3·13-s + 2.88e3·15-s − 3.14e4·17-s + 1.99e4·19-s + 6.17e3·21-s + 7.71e4·23-s − 5.25e4·25-s − 7.29e4·27-s − 1.93e5·29-s + 2.63e4·31-s − 1.02e5·33-s + 5.48e4·35-s + 2.04e5·37-s + 2.49e4·39-s − 6.63e5·41-s + 3.35e5·43-s − 2.98e5·45-s − 1.11e6·47-s + 1.17e5·49-s − 5.65e5·51-s + 1.12e5·53-s − 9.12e5·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 0.572·5-s + 0.377·7-s − 0.851·9-s − 1.29·11-s + 0.175·13-s + 0.220·15-s − 1.55·17-s + 0.667·19-s + 0.145·21-s + 1.32·23-s − 0.672·25-s − 0.712·27-s − 1.47·29-s + 0.158·31-s − 0.497·33-s + 0.216·35-s + 0.663·37-s + 0.0674·39-s − 1.50·41-s + 0.644·43-s − 0.487·45-s − 1.57·47-s + 1/7·49-s − 0.597·51-s + 0.104·53-s − 0.739·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 - 2 p^{2} T + p^{7} T^{2} \) |
| 5 | \( 1 - 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 + 5704 T + p^{7} T^{2} \) |
| 13 | \( 1 - 1388 T + p^{7} T^{2} \) |
| 17 | \( 1 + 31434 T + p^{7} T^{2} \) |
| 19 | \( 1 - 19966 T + p^{7} T^{2} \) |
| 23 | \( 1 - 77136 T + p^{7} T^{2} \) |
| 29 | \( 1 + 193374 T + p^{7} T^{2} \) |
| 31 | \( 1 - 26356 T + p^{7} T^{2} \) |
| 37 | \( 1 - 204346 T + p^{7} T^{2} \) |
| 41 | \( 1 + 663050 T + p^{7} T^{2} \) |
| 43 | \( 1 - 335920 T + p^{7} T^{2} \) |
| 47 | \( 1 + 1119812 T + p^{7} T^{2} \) |
| 53 | \( 1 - 112782 T + p^{7} T^{2} \) |
| 59 | \( 1 + 536154 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1170264 T + p^{7} T^{2} \) |
| 67 | \( 1 + 3890660 T + p^{7} T^{2} \) |
| 71 | \( 1 + 2505344 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1435070 T + p^{7} T^{2} \) |
| 79 | \( 1 + 176536 T + p^{7} T^{2} \) |
| 83 | \( 1 - 6211622 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4729062 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2129562 T + p^{7} T^{2} \) |
| 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
−11.53371819450485578904168707737, −10.76381923794521689396849579527, −9.453415827882679999475134433158, −8.529686728051361656468173591978, −7.42872163831214295384204678376, −5.92401211239699199959893078217, −4.87863407365946872947735746375, −3.07101825806184486729329248956, −1.95178367957386716520545544178, 0,
1.95178367957386716520545544178, 3.07101825806184486729329248956, 4.87863407365946872947735746375, 5.92401211239699199959893078217, 7.42872163831214295384204678376, 8.529686728051361656468173591978, 9.453415827882679999475134433158, 10.76381923794521689396849579527, 11.53371819450485578904168707737
