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 | \( 1 + (-8.55e3 - 8.55e3i)T + 8.85e7iT^{2} \) |
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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
−13.74129327489472865246681997862, −12.55405301488950536203136524040, −11.33074331448218221021051120149, −10.14341723917353086614511595025, −8.503506752713963402867187098384, −7.64718106813488991655038210042, −6.59453978076945189708970205520, −5.40502016018711401965756541817, −2.45777318112433199143028461860, −0.76578567989378638861092152207,
2.01903410027451328849633560125, 4.18808379457413530833787960209, 5.05497803725509012840740769734, 7.30482887869905854809171451464, 8.974415434692646879142326846420, 9.597581784163030753516922524639, 10.55281544743404011233038173518, 11.65146014319163759982367065427, 12.79810947069668887105006603298, 14.38767916124205121306494080633