| L(s) = 1 | + (−3.36 + 10.8i)2-s + (10.2 − 24.7i)3-s + (−105. − 72.7i)4-s + (148. − 61.4i)5-s + (232. + 194. i)6-s + (−729. + 729. i)7-s + (1.14e3 − 892. i)8-s + (1.03e3 + 1.03e3i)9-s + (164. + 1.80e3i)10-s + (3.09e3 + 7.47e3i)11-s + (−2.87e3 + 1.86e3i)12-s + (−9.28e3 − 3.84e3i)13-s + (−5.42e3 − 1.03e4i)14-s − 4.29e3i·15-s + (5.80e3 + 1.53e4i)16-s + 2.04e4i·17-s + ⋯ |
| L(s) = 1 | + (−0.297 + 0.954i)2-s + (0.219 − 0.529i)3-s + (−0.822 − 0.568i)4-s + (0.530 − 0.219i)5-s + (0.439 + 0.366i)6-s + (−0.804 + 0.804i)7-s + (0.787 − 0.616i)8-s + (0.475 + 0.475i)9-s + (0.0519 + 0.572i)10-s + (0.701 + 1.69i)11-s + (−0.480 + 0.310i)12-s + (−1.17 − 0.485i)13-s + (−0.528 − 1.00i)14-s − 0.328i·15-s + (0.354 + 0.935i)16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.538258 + 1.08378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.538258 + 1.08378i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.36 - 10.8i)T \) |
| good | 3 | \( 1 + (-10.2 + 24.7i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-148. + 61.4i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (729. - 729. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-3.09e3 - 7.47e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (9.28e3 + 3.84e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 2.04e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (3.77e4 + 1.56e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-7.81e4 - 7.81e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (-1.59e4 + 3.84e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.15e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-6.94e4 + 2.87e4i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (1.91e5 + 1.91e5i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-1.94e5 - 4.69e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.13e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (5.61e5 + 1.35e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (1.22e6 - 5.08e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-7.37e5 + 1.77e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (3.16e5 - 7.64e5i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-7.50e5 + 7.50e5i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (2.59e6 + 2.59e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 2.61e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-4.42e6 - 1.83e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-6.76e6 + 6.76e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 8.52e6T + 8.07e13T^{2} \) |
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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
−15.46896976543450453295776033704, −14.83455582395417674777059851814, −13.15045578515759812372911219698, −12.59690464315480560247092579467, −10.03033170069512203773160598379, −9.195310673204480764711233198772, −7.55874976357293265361888261487, −6.44821618386409204971537772283, −4.83223623507634184396303178429, −1.87074660459927501560044218883,
0.62738857578303034905934447948, 2.93571787774331046241587729667, 4.29587693151659254361085025223, 6.70159160459572244451192070362, 8.829985773909236692752021122231, 9.825875052835356457970560136491, 10.74725392688421872154685428013, 12.26193895957056911244209167161, 13.56413825118459099963397035301, 14.47825604985698778675317872486
