| L(s) = 1 | + (3.23 + 2.35i)2-s + (2.78 + 8.55i)3-s + (4.94 + 15.2i)4-s + (54.0 − 14.4i)5-s + (−11.1 + 34.2i)6-s + 182.·7-s + (−19.7 + 60.8i)8-s + (−65.5 + 47.6i)9-s + (208. + 80.3i)10-s + (0.112 + 0.0816i)11-s + (−116. + 84.6i)12-s + (476. − 345. i)13-s + (589. + 427. i)14-s + (273. + 422. i)15-s + (−207. + 150. i)16-s + (214. − 661. i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.966 − 0.257i)5-s + (−0.126 + 0.388i)6-s + 1.40·7-s + (−0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.659 + 0.254i)10-s + (0.000280 + 0.000203i)11-s + (−0.233 + 0.169i)12-s + (0.781 − 0.567i)13-s + (0.803 + 0.583i)14-s + (0.313 + 0.484i)15-s + (−0.202 + 0.146i)16-s + (0.180 − 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.974785955\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.974785955\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 + (-2.78 - 8.55i)T \) |
| 5 | \( 1 + (-54.0 + 14.4i)T \) |
| good | 7 | \( 1 - 182.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-0.112 - 0.0816i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-476. + 345. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-214. + 661. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-436. + 1.34e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (584. + 424. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-1.77e3 - 5.45e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.23e3 - 6.86e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.10e4 - 8.03e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (8.25e3 - 5.99e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 112.T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.79e3 + 5.51e3i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.50e3 + 4.62e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.02e4 - 7.46e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (7.39e3 + 5.37e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.69e4 + 5.22e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.91e3 + 5.89e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-1.62e4 - 1.18e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.42e3 - 7.45e3i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (4.50e3 - 1.38e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (5.27e4 + 3.82e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-4.20e4 - 1.29e5i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−12.39490382287157814557078765458, −11.21365871250775224777315593891, −10.35432195881617266374537449465, −8.958033269583717518807414284111, −8.220210478257074781900405349696, −6.77368176198649557563837417740, −5.30490248148513194712667281316, −4.85237116173613563751311548391, −3.13801110937675970076094587421, −1.51951189416126750017657164610,
1.40472752880750452959824524037, 2.15532433274302920080428913320, 3.90642481888532292535168058391, 5.39504888147902505922116850229, 6.27929765119585152283565992047, 7.68547170406803053268406876875, 8.819878870042132009832480610794, 10.11685990746156631669007857572, 11.11754008718341809124612380621, 11.92926187634474630168697267802
