| L(s) = 1 | + 53.9·5-s − 188. i·7-s − 770. i·11-s − 670.·13-s − 591.·17-s + 9.39i·19-s + 1.42e4i·23-s − 1.27e4·25-s − 3.37e4·29-s − 2.43e4i·31-s − 1.01e4i·35-s + 6.34e4·37-s + 720.·41-s + 1.29e5i·43-s + 5.34e4i·47-s + ⋯ |
| L(s) = 1 | + 0.431·5-s − 0.548i·7-s − 0.579i·11-s − 0.305·13-s − 0.120·17-s + 0.00137i·19-s + 1.17i·23-s − 0.813·25-s − 1.38·29-s − 0.817i·31-s − 0.236i·35-s + 1.25·37-s + 0.0104·41-s + 1.63i·43-s + 0.515i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.222820293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222820293\) |
| \(L(4)\) |
|
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 | \( 1 - 53.9T + 1.56e4T^{2} \) |
| 7 | \( 1 + 188. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 770. iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 670.T + 4.82e6T^{2} \) |
| 17 | \( 1 + 591.T + 2.41e7T^{2} \) |
| 19 | \( 1 - 9.39iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.42e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 3.37e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.43e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.34e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 720.T + 4.75e9T^{2} \) |
| 43 | \( 1 - 1.29e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 5.34e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 1.99e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 4.25e3iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.18e4T + 5.15e10T^{2} \) |
| 67 | \( 1 - 8.32e4iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.16e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 7.47e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.12e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 7.61e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 1.08e6T + 4.96e11T^{2} \) |
| 97 | \( 1 - 1.37e6T + 8.32e11T^{2} \) |
| 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
−9.718204760075506715019853327970, −9.432663655238153738148675181145, −8.058084091221810469019948630994, −7.44638473790513663187136094419, −6.26091707511258126460337375462, −5.54255694746871819793502872543, −4.34379594124700668244906031210, −3.36462981548989046892727240201, −2.12770106066536740963830636431, −0.989051653864485370674605467084,
0.25827817000743670942041260036, 1.77971107389034633267279491249, 2.56019243936173421014232521547, 3.91053491087410897421342602189, 5.00297021486818400328853399742, 5.86437940351365058614881372652, 6.81663146971894444128063464965, 7.76769398309789129793329975865, 8.789099326601137119677174901304, 9.534561249671924470451130420994
