| L(s) = 1 | + (−4.14 − 3.13i)3-s + (−4.27 − 1.55i)5-s + (−4.69 + 26.6i)7-s + (7.28 + 25.9i)9-s + (50.9 − 18.5i)11-s + (42.8 − 35.9i)13-s + (12.8 + 19.8i)15-s + (49.3 + 85.5i)17-s + (−68.0 + 117. i)19-s + (103. − 95.5i)21-s + (25.4 + 144. i)23-s + (−79.9 − 67.0i)25-s + (51.4 − 130. i)27-s + (−37.6 − 31.5i)29-s + (26.6 + 150. i)31-s + ⋯ |
| L(s) = 1 | + (−0.796 − 0.604i)3-s + (−0.382 − 0.139i)5-s + (−0.253 + 1.43i)7-s + (0.269 + 0.962i)9-s + (1.39 − 0.508i)11-s + (0.913 − 0.766i)13-s + (0.220 + 0.341i)15-s + (0.704 + 1.22i)17-s + (−0.822 + 1.42i)19-s + (1.07 − 0.992i)21-s + (0.230 + 1.30i)23-s + (−0.639 − 0.536i)25-s + (0.366 − 0.930i)27-s + (−0.240 − 0.202i)29-s + (0.154 + 0.874i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.705 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.990584 + 0.411895i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990584 + 0.411895i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.14 + 3.13i)T \) |
| good | 5 | \( 1 + (4.27 + 1.55i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (4.69 - 26.6i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (-50.9 + 18.5i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (-42.8 + 35.9i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-49.3 - 85.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (68.0 - 117. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-25.4 - 144. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (37.6 + 31.5i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-26.6 - 150. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-11.7 - 20.3i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-317. + 266. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (269. - 98.2i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (48.6 - 275. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 248.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-243. - 88.6i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (57.5 - 326. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-71.5 + 60.0i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (333. + 577. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (106. - 185. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (421. + 353. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-310. - 260. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-504. + 874. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (27.6 - 10.0i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−13.00167312774931002304380208953, −12.20034440857848220386421432109, −11.59261371077193543361327472309, −10.35739669925446541144582478668, −8.818627818750060841173289446870, −7.938036770981701143748385237289, −6.14088820898867858768153422699, −5.78768282725026002080884208857, −3.71267347893884031695742141394, −1.50355209663825879193391569368,
0.74176657499407518577173775132, 3.79980303164887703895843553897, 4.55785041936407963382192483376, 6.48670940374121529887085581683, 7.13313614846294702381410598928, 9.043804533206665273958646913505, 9.981323650267909769036825854502, 11.15112870772688772720654974999, 11.66702009090089843503803266445, 13.07480066228841037282067739709
