L(s) = 1 | + (−1 − 1.73i)2-s + (4.69 − 8.12i)3-s + (−1.99 + 3.46i)4-s + (−4.69 − 8.12i)5-s − 18.7·6-s + 7.99·8-s + (−30.5 − 52.8i)9-s + (−9.38 + 16.2i)10-s + (−10 + 17.3i)11-s + (18.7 + 32.4i)12-s + 65.6·13-s − 88.0·15-s + (−8 − 13.8i)16-s + (−28.1 + 48.7i)17-s + (−61 + 105. i)18-s + (−4.69 − 8.12i)19-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.902 − 1.56i)3-s + (−0.249 + 0.433i)4-s + (−0.419 − 0.726i)5-s − 1.27·6-s + 0.353·8-s + (−1.12 − 1.95i)9-s + (−0.296 + 0.513i)10-s + (−0.274 + 0.474i)11-s + (0.451 + 0.781i)12-s + 1.40·13-s − 1.51·15-s + (−0.125 − 0.216i)16-s + (−0.401 + 0.695i)17-s + (−0.798 + 1.38i)18-s + (−0.0566 − 0.0980i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0915558 - 1.44272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0915558 - 1.44272i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-4.69 + 8.12i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (4.69 + 8.12i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (10 - 17.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 65.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (28.1 - 48.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (4.69 + 8.12i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (24 + 41.5i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-103. + 178. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-39 - 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436T + 7.95e4T^{2} \) |
| 47 | \( 1 + (103. + 178. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (31 - 53.6i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-333. + 576. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (136. + 235. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (290 - 502. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 544T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-300. + 519. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-340 - 588. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 196.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-750. - 1.29e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 656.T + 9.12e5T^{2} \) |
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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
−12.92623340602245963381738890110, −12.17367727827051415933211004844, −11.01736482175531653562721628363, −9.253498817837874511772572686556, −8.386520812354337881093607107443, −7.70781109034931983143823401402, −6.27285004242027547278664608027, −3.94049487770476489304774672973, −2.25659947204738864328203188092, −0.893627361577646493719601242722,
3.08169621306801976619009874574, 4.23185301754449447060188513828, 5.77558088722491183261578313606, 7.50743599629853213524003382053, 8.628291730027317161880456142578, 9.386772777887015292392450120436, 10.65732476846797292052499693117, 11.15037834653918480614456127576, 13.46789709666785119542865065822, 14.28591323521111707633396623002