L(s) = 1 | + (1.46 − 1.36i)2-s + (8.68 − 1.30i)3-s + (0.298 − 3.98i)4-s + (−3.71 + 9.46i)5-s + (10.9 − 13.7i)6-s + (15.2 + 10.4i)7-s + (−4.98 − 6.25i)8-s + (47.9 − 14.7i)9-s + (7.42 + 18.9i)10-s + (−17.7 − 5.47i)11-s + (−2.62 − 35.0i)12-s + (−10.3 − 45.4i)13-s + (36.6 − 5.44i)14-s + (−19.8 + 87.0i)15-s + (−15.8 − 2.38i)16-s + (−100. + 68.6i)17-s + ⋯ |
L(s) = 1 | + (0.518 − 0.480i)2-s + (1.67 − 0.251i)3-s + (0.0373 − 0.498i)4-s + (−0.332 + 0.846i)5-s + (0.745 − 0.934i)6-s + (0.825 + 0.564i)7-s + (−0.220 − 0.276i)8-s + (1.77 − 0.547i)9-s + (0.234 + 0.598i)10-s + (−0.486 − 0.149i)11-s + (−0.0631 − 0.842i)12-s + (−0.221 − 0.970i)13-s + (0.699 − 0.103i)14-s + (−0.342 + 1.49i)15-s + (−0.247 − 0.0372i)16-s + (−1.43 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.847 + 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.04948 - 0.876296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.04948 - 0.876296i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.46 + 1.36i)T \) |
| 7 | \( 1 + (-15.2 - 10.4i)T \) |
good | 3 | \( 1 + (-8.68 + 1.30i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (3.71 - 9.46i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (17.7 + 5.47i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (10.3 + 45.4i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (100. - 68.6i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (45.0 + 78.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (101. + 68.9i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-226. - 108. i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-52.3 + 90.6i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-12.0 - 160. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (26.1 + 32.7i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (153. - 192. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (150. - 140. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (-29.4 + 392. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-165. - 422. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-16.2 - 216. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-393. + 682. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-931. + 448. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (283. + 263. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (537. + 930. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (215. - 944. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (1.37e3 - 425. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−13.42375994658354274141829820506, −12.60221844901904213499731079235, −11.16452878803876886858506125233, −10.23755265818702962075130105821, −8.691583633153228562723133609395, −8.029757834315263617209778465868, −6.59973043754181366709298404195, −4.56493090989668810412239287346, −3.05239743053364496861716834374, −2.19838953407190441663813879971,
2.18733568440490556829112352857, 4.05091359807047241950210102026, 4.73183774533207882650534046591, 7.04064276411111855347984678788, 8.159772148707891909402172713963, 8.693332119240823491999466601852, 9.983574264909082308467867839131, 11.64906159900648961774822954255, 12.88664183048999338043754466554, 13.95331591246358005227164921337