| L(s) = 1 | + (2.68 − 0.874i)2-s + (5.70 + 4.14i)3-s + (6.47 − 4.70i)4-s + (−0.903 + 2.77i)5-s + (18.9 + 6.16i)6-s + (−3.94 − 5.43i)7-s + (13.3 − 18.3i)8-s + (−9.66 − 29.7i)9-s + 8.26i·10-s + (−100. + 67.8i)11-s + 56.4·12-s + (−149. + 48.4i)13-s + (−15.3 − 11.1i)14-s + (−16.6 + 12.1i)15-s + (19.7 − 60.8i)16-s + (−72.6 − 23.6i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (0.633 + 0.460i)3-s + (0.404 − 0.293i)4-s + (−0.0361 + 0.111i)5-s + (0.526 + 0.171i)6-s + (−0.0805 − 0.110i)7-s + (0.207 − 0.286i)8-s + (−0.119 − 0.367i)9-s + 0.0826i·10-s + (−0.827 + 0.560i)11-s + 0.391·12-s + (−0.883 + 0.286i)13-s + (−0.0784 − 0.0569i)14-s + (−0.0740 + 0.0538i)15-s + (0.0772 − 0.237i)16-s + (−0.251 − 0.0816i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0109i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.99242 - 0.0109013i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99242 - 0.0109013i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.68 + 0.874i)T \) |
| 11 | \( 1 + (100. - 67.8i)T \) |
| good | 3 | \( 1 + (-5.70 - 4.14i)T + (25.0 + 77.0i)T^{2} \) |
| 5 | \( 1 + (0.903 - 2.77i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (3.94 + 5.43i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (149. - 48.4i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (72.6 + 23.6i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-96.2 + 132. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 803.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-352. - 485. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (169. + 522. i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (1.28e3 - 931. i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (1.47e3 - 2.02e3i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 2.74e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-3.05e3 - 2.21e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-796. - 2.45e3i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (1.32e3 - 965. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (3.90e3 + 1.26e3i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 - 1.28e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (-470. + 1.44e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (4.72e3 + 6.50e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (-1.64e3 + 534. i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (7.88e3 + 2.56e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 - 9.87e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-322. - 992. i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−17.09834044267755371457144852005, −15.46676488021944148562721014066, −14.80047318405576243511786646657, −13.46763090118870937215032636832, −12.14843470863451345663606174038, −10.52848044933327671603058093920, −9.142206459049185939280561847958, −7.10157212640951503288585260378, −4.85877282448112001423328539298, −2.95344681081298942812920248758,
2.77256955430456118478933495881, 5.21301964670799711798814230634, 7.24951733985135484515576123637, 8.563869735505342867617394773310, 10.69153536086337538330980841347, 12.42764316795933786625145632105, 13.45009705717544816027638579145, 14.50521374850475570150333317543, 15.78089387059040226509698835897, 17.05642977056289636054768855214
