| L(s) = 1 | + (0.347 − 1.96i)2-s + (2.29 + 1.92i)3-s + (−3.75 − 1.36i)4-s + (7.64 − 2.78i)5-s + (4.59 − 3.85i)6-s + (15.4 + 26.8i)7-s + (−4 + 6.92i)8-s + (1.56 + 8.86i)9-s + (−2.82 − 16.0i)10-s + (9.20 − 15.9i)11-s + (−6 − 10.3i)12-s + (9.78 − 8.20i)13-s + (58.2 − 21.1i)14-s + (22.9 + 8.35i)15-s + (12.2 + 10.2i)16-s + (6.17 − 34.9i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (0.684 − 0.249i)5-s + (0.312 − 0.262i)6-s + (0.836 + 1.44i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.0894 − 0.507i)10-s + (0.252 − 0.436i)11-s + (−0.144 − 0.249i)12-s + (0.208 − 0.175i)13-s + (1.11 − 0.404i)14-s + (0.395 + 0.143i)15-s + (0.191 + 0.160i)16-s + (0.0880 − 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.20826 - 0.312535i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.20826 - 0.312535i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.347 + 1.96i)T \) |
| 3 | \( 1 + (-2.29 - 1.92i)T \) |
| 19 | \( 1 + (-82.5 + 6.92i)T \) |
| good | 5 | \( 1 + (-7.64 + 2.78i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-15.4 - 26.8i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-9.20 + 15.9i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-9.78 + 8.20i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-6.17 + 34.9i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (-75.4 - 27.4i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (23.9 + 135. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (83.8 + 145. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 300.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-262. - 220. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (449. - 163. i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (-12.6 - 71.6i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (435. + 158. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (56.7 - 321. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (25.7 + 9.36i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (154. + 876. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-382. + 139. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-242. - 203. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (-90.9 - 76.2i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (640. + 1.10e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-31.2 + 26.2i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-156. + 886. i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−13.11335081857841916852491245454, −11.85094449556122633417525011795, −11.17107014338256567437029042232, −9.650780334137598648466479899955, −9.086418423223030518594503544661, −7.996557910794114162936774542192, −5.80322925650657317472538274167, −4.93972683867353538984975651276, −3.10094780579570374650311705907, −1.75286864516471748463411595389,
1.46649940577381504447608441330, 3.70949586733945837821853319691, 5.14298681729477230786337418643, 6.74255494425900599341909494193, 7.44486856936964511094362196406, 8.619535355812598852094033981467, 9.896277759038028216933707846260, 10.90885170302989152701373557214, 12.41863925391147193472992991961, 13.62257992133835779901082168718
