| L(s) = 1 | + (3.79 + 3.54i)3-s + (5.15 + 8.93i)5-s + (−5.91 − 17.5i)7-s + (1.80 + 26.9i)9-s + (−2.90 − 1.67i)11-s + (73.0 + 42.1i)13-s + (−12.1 + 52.1i)15-s + (37.1 + 64.3i)17-s + (7.67 + 4.43i)19-s + (39.8 − 87.6i)21-s + (−119. + 68.9i)23-s + (9.33 − 16.1i)25-s + (−88.7 + 108. i)27-s + (−127. + 73.8i)29-s − 37.9i·31-s + ⋯ |
| L(s) = 1 | + (0.730 + 0.683i)3-s + (0.461 + 0.798i)5-s + (−0.319 − 0.947i)7-s + (0.0669 + 0.997i)9-s + (−0.0796 − 0.0459i)11-s + (1.55 + 0.900i)13-s + (−0.208 + 0.898i)15-s + (0.530 + 0.918i)17-s + (0.0926 + 0.0535i)19-s + (0.413 − 0.910i)21-s + (−1.08 + 0.625i)23-s + (0.0746 − 0.129i)25-s + (−0.632 + 0.774i)27-s + (−0.818 + 0.472i)29-s − 0.219i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0836 - 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0836 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.423450469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.423450469\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.79 - 3.54i)T \) |
| 7 | \( 1 + (5.91 + 17.5i)T \) |
| good | 5 | \( 1 + (-5.15 - 8.93i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (2.90 + 1.67i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-73.0 - 42.1i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-37.1 - 64.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-7.67 - 4.43i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (119. - 68.9i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (127. - 73.8i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + 37.9iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (35.7 - 61.9i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (98.5 - 170. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (102. + 177. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 - 362.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-586. + 338. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 - 564.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 190. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 856.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 53.7iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-234. + 135. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 - 520.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-66.9 - 115. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-417. + 722. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-591. + 341. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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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
−11.52887341190092938769780900094, −10.54131248722261735566489428366, −10.12153814828089011907001183101, −9.011301048848685256064434797629, −7.999625003819961994967135146696, −6.85822286857206384865703075813, −5.80358560721009176051402334273, −4.05948531518261630050807288097, −3.42981345890549832149102080729, −1.78473891472336394976016421597,
0.942117586408419431727901404650, 2.35456292820634982718909941735, 3.62742313788873714452301709117, 5.44342433132045086886799538437, 6.18378693198327443907681479965, 7.60335564446914112996153976906, 8.629141707078081502640612894058, 9.099353151829034765145462637745, 10.20834798540477628738437175850, 11.69664489768475373697499440828
