L(s) = 1 | + (−5.00 + 8.66i)5-s + (1.56 + 18.4i)7-s + (8.94 − 5.16i)11-s + 52.4i·13-s + (0.584 + 1.01i)17-s + (−86.7 − 50.0i)19-s + (90.1 + 52.0i)23-s + (12.4 + 21.6i)25-s − 187. i·29-s + (−107. + 62.0i)31-s + (−167. − 78.6i)35-s + (16.0 − 27.7i)37-s − 415.·41-s − 193.·43-s + (196. − 340. i)47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.774i)5-s + (0.0847 + 0.996i)7-s + (0.245 − 0.141i)11-s + 1.11i·13-s + (0.00833 + 0.0144i)17-s + (−1.04 − 0.604i)19-s + (0.817 + 0.471i)23-s + (0.0999 + 0.173i)25-s − 1.20i·29-s + (−0.622 + 0.359i)31-s + (−0.809 − 0.379i)35-s + (0.0712 − 0.123i)37-s − 1.58·41-s − 0.685·43-s + (0.609 − 1.05i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0499i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7339193204\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7339193204\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.56 - 18.4i)T \) |
good | 5 | \( 1 + (5.00 - 8.66i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-8.94 + 5.16i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 52.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-0.584 - 1.01i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (86.7 + 50.0i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-90.1 - 52.0i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 187. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (107. - 62.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-16.0 + 27.7i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 415.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 193.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-196. + 340. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (74.7 - 43.1i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (102. + 177. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (183. + 105. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-364. - 631. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 315. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (899. - 519. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (607. - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 333.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (168. - 291. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 893. iT - 9.12e5T^{2} \) |
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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
−11.30271997774197777747266124048, −10.07620292212415076991650751657, −9.053938172978986607746506097305, −8.441567392362358909741184793373, −7.14118810871221284163893507231, −6.53601650154206756270443896498, −5.37730090003196177860065031478, −4.18588966736660344170874358201, −3.02081474055623534268483788458, −1.88165550335170373594385424248,
0.22978051100743609279816367033, 1.41445269370377873885897441507, 3.23446215565601303013513829811, 4.30156042422513625662230279639, 5.12243768691317411470043933160, 6.42273693089441043200024791670, 7.44128060558661607863591825772, 8.256966259074254325712292722374, 9.018049315433014408448458142563, 10.29566540307007925687954716044