L(s) = 1 | + (−2 − 2i)2-s + 12.3i·3-s + 8i·4-s + (10.7 − 10.7i)5-s + (24.6 − 24.6i)6-s + 35.0·7-s + (16 − 16i)8-s − 70.4·9-s − 43.0·10-s + 142. i·11-s − 98.4·12-s + (−181. + 181. i)13-s + (−70.1 − 70.1i)14-s + (132. + 132. i)15-s − 64·16-s + (−88.0 + 88.0i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + 1.36i·3-s + 0.5i·4-s + (0.430 − 0.430i)5-s + (0.683 − 0.683i)6-s + 0.715·7-s + (0.250 − 0.250i)8-s − 0.870·9-s − 0.430·10-s + 1.17i·11-s − 0.683·12-s + (−1.07 + 1.07i)13-s + (−0.357 − 0.357i)14-s + (0.588 + 0.588i)15-s − 0.250·16-s + (−0.304 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0880 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.831535 + 0.908280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.831535 + 0.908280i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2 + 2i)T \) |
| 37 | \( 1 + (7.65 + 1.36e3i)T \) |
good | 3 | \( 1 - 12.3iT - 81T^{2} \) |
| 5 | \( 1 + (-10.7 + 10.7i)T - 625iT^{2} \) |
| 7 | \( 1 - 35.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 142. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (181. - 181. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (88.0 - 88.0i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + (-13.8 + 13.8i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + (-79.7 + 79.7i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + (-420. - 420. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + (624. + 624. i)T + 9.23e5iT^{2} \) |
| 41 | \( 1 - 1.23e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-607. + 607. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 - 2.97e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.04e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.89e3 + 1.89e3i)T - 1.21e7iT^{2} \) |
| 61 | \( 1 + (-4.01e3 - 4.01e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 - 2.33e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 3.64e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 9.94e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-6.40e3 + 6.40e3i)T - 3.89e7iT^{2} \) |
| 83 | \( 1 + 7.15e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-2.25e3 - 2.25e3i)T + 6.27e7iT^{2} \) |
| 97 | \( 1 + (-8.55e3 + 8.55e3i)T - 8.85e7iT^{2} \) |
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
−14.38767916124205121306494080633, −12.79810947069668887105006603298, −11.65146014319163759982367065427, −10.55281544743404011233038173518, −9.597581784163030753516922524639, −8.974415434692646879142326846420, −7.30482887869905854809171451464, −5.05497803725509012840740769734, −4.18808379457413530833787960209, −2.01903410027451328849633560125,
0.76578567989378638861092152207, 2.45777318112433199143028461860, 5.40502016018711401965756541817, 6.59453978076945189708970205520, 7.64718106813488991655038210042, 8.503506752713963402867187098384, 10.14341723917353086614511595025, 11.33074331448218221021051120149, 12.55405301488950536203136524040, 13.74129327489472865246681997862