| L(s) = 1 | + (4.97 + 1.49i)3-s + (−0.619 + 1.07i)5-s + (−8.82 − 16.2i)7-s + (22.5 + 14.9i)9-s + (−56.1 + 32.4i)11-s + 54.9i·13-s + (−4.69 + 4.41i)15-s + (47.9 + 82.9i)17-s + (−23.2 − 13.4i)19-s + (−19.4 − 94.2i)21-s + (150. + 86.6i)23-s + (61.7 + 106. i)25-s + (89.6 + 107. i)27-s + 16.9i·29-s + (−66.1 + 38.1i)31-s + ⋯ |
| L(s) = 1 | + (0.957 + 0.288i)3-s + (−0.0554 + 0.0959i)5-s + (−0.476 − 0.879i)7-s + (0.833 + 0.552i)9-s + (−1.53 + 0.888i)11-s + 1.17i·13-s + (−0.0807 + 0.0759i)15-s + (0.683 + 1.18i)17-s + (−0.281 − 0.162i)19-s + (−0.202 − 0.979i)21-s + (1.36 + 0.785i)23-s + (0.493 + 0.855i)25-s + (0.638 + 0.769i)27-s + 0.108i·29-s + (−0.383 + 0.221i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0770 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0770 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.942745800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942745800\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.97 - 1.49i)T \) |
| 7 | \( 1 + (8.82 + 16.2i)T \) |
| good | 5 | \( 1 + (0.619 - 1.07i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (56.1 - 32.4i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 54.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-47.9 - 82.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (23.2 + 13.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-150. - 86.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 16.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (66.1 - 38.1i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-69.3 + 120. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 100.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 197.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-161. + 280. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-248. + 143. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-84.2 - 145. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (166. + 96.1i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (442. + 765. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 661. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-125. + 72.2i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (228. - 395. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 967.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (605. - 1.04e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 777. iT - 9.12e5T^{2} \) |
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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
−11.04946512561731049480816123286, −10.31317405797908233627686896516, −9.594560756585177885428982773468, −8.587912731790632977019895843522, −7.47126948216938048221669506320, −6.97577062158918917811006285128, −5.20802113180185061405548808522, −4.11519894910420539082161355766, −3.09352736597239194338829969736, −1.70515644110826213685323374163,
0.60472237426237063462739737041, 2.73533972315050766053938166798, 3.02962311354107885460866945459, 4.94400273811267474048264252647, 5.93915738085901648565935313706, 7.26553273480599174587306975150, 8.204129399723900109320143913770, 8.787258751013922695227075748481, 9.895356252568164007402147038194, 10.69846324779681479345364843844
