| L(s) = 1 | + (−2.90 − 2.90i)3-s + (59.6 − 109. i)5-s + (236. − 236. i)7-s − 712. i·9-s + 564.·11-s + (134. + 134. i)13-s + (−493. + 145. i)15-s + (−4.79e3 + 4.79e3i)17-s + 3.93e3i·19-s − 1.37e3·21-s + (7.89e3 + 7.89e3i)23-s + (−8.49e3 − 1.31e4i)25-s + (−4.19e3 + 4.19e3i)27-s − 3.31e4i·29-s + 5.47e4·31-s + ⋯ |
| L(s) = 1 | + (−0.107 − 0.107i)3-s + (0.477 − 0.878i)5-s + (0.688 − 0.688i)7-s − 0.976i·9-s + 0.423·11-s + (0.0614 + 0.0614i)13-s + (−0.146 + 0.0432i)15-s + (−0.976 + 0.976i)17-s + 0.573i·19-s − 0.148·21-s + (0.648 + 0.648i)23-s + (−0.543 − 0.839i)25-s + (−0.213 + 0.213i)27-s − 1.35i·29-s + 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.37549 - 0.797987i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37549 - 0.797987i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-59.6 + 109. i)T \) |
| good | 3 | \( 1 + (2.90 + 2.90i)T + 729iT^{2} \) |
| 7 | \( 1 + (-236. + 236. i)T - 1.17e5iT^{2} \) |
| 11 | \( 1 - 564.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-134. - 134. i)T + 4.82e6iT^{2} \) |
| 17 | \( 1 + (4.79e3 - 4.79e3i)T - 2.41e7iT^{2} \) |
| 19 | \( 1 - 3.93e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-7.89e3 - 7.89e3i)T + 1.48e8iT^{2} \) |
| 29 | \( 1 + 3.31e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 5.47e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + (4.23e4 - 4.23e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 - 2.80e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + (-8.40e4 - 8.40e4i)T + 6.32e9iT^{2} \) |
| 47 | \( 1 + (1.23e5 - 1.23e5i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 + (-2.52e4 - 2.52e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + 1.84e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 2.69e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-1.74e5 + 1.74e5i)T - 9.04e10iT^{2} \) |
| 71 | \( 1 + 3.67e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.96e5 - 2.96e5i)T + 1.51e11iT^{2} \) |
| 79 | \( 1 - 7.65e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (5.99e5 + 5.99e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 2.10e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-9.84e5 + 9.84e5i)T - 8.32e11iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.24518001333144540452451568428, −15.50649809653027864419007208890, −14.07965896268492697537787954010, −12.83678462119372639941390403638, −11.47792516729596935055313303508, −9.746114906766019797933329234106, −8.305285738688042621119506340722, −6.29935567441858563326282981443, −4.34391199346695927959342478127, −1.22137748304524717948015388435,
2.39098729029253178338195347050, 5.09272774833120815878555134405, 6.93516771446006128863607017933, 8.801626797419643690854203862082, 10.54238763652132318779186025816, 11.60060673996484011226122050652, 13.49506880531757893715414591759, 14.58834012512545011262688046366, 15.83894372155390685439444356178, 17.40669476523184394661218063329
