| L(s) = 1 | + (−2.5 − 4.33i)3-s + (2.5 − 4.33i)5-s + (0.999 − 1.73i)9-s + (7.5 + 12.9i)11-s + 13·13-s − 25.0·15-s + (−13.5 − 23.3i)17-s + (−77 + 133. i)19-s + (93 − 161. i)23-s + (−12.5 − 21.6i)25-s − 144.·27-s + 3·29-s + (−164 − 284. i)31-s + (37.5 − 64.9i)33-s + (−127 + 219. i)37-s + ⋯ |
| L(s) = 1 | + (−0.481 − 0.833i)3-s + (0.223 − 0.387i)5-s + (0.0370 − 0.0641i)9-s + (0.205 + 0.356i)11-s + 0.277·13-s − 0.430·15-s + (−0.192 − 0.333i)17-s + (−0.929 + 1.61i)19-s + (0.843 − 1.46i)23-s + (−0.100 − 0.173i)25-s − 1.03·27-s + 0.0192·29-s + (−0.950 − 1.64i)31-s + (0.197 − 0.342i)33-s + (−0.564 + 0.977i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3267582227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3267582227\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (2.5 + 4.33i)T + (-13.5 + 23.3i)T^{2} \) |
| 11 | \( 1 + (-7.5 - 12.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 13T + 2.19e3T^{2} \) |
| 17 | \( 1 + (13.5 + 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (77 - 133. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-93 + 161. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 3T + 2.43e4T^{2} \) |
| 31 | \( 1 + (164 + 284. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (127 - 219. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 96T + 6.89e4T^{2} \) |
| 43 | \( 1 - 134T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-25.5 + 44.1i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (120 + 207. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (198 + 342. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (308 - 533. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (148 + 256. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 48T + 3.57e5T^{2} \) |
| 73 | \( 1 + (161 + 278. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (329.5 - 570. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 300T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-510 + 883. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 199T + 9.12e5T^{2} \) |
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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
−9.045421544376108222864052123903, −8.207718243924617072248016641471, −7.31672064278767358603216596795, −6.43141615348541834869962123388, −5.90103358176555698201045829837, −4.72546423493906028613694773667, −3.75431605309231219861817224812, −2.20627018580590331258230655295, −1.27327542420351744444946177677, −0.088561251189226392705214232137,
1.61232492289303765167476126310, 2.99959866714197180010797992242, 3.99927793883202316040319704788, 4.96496748883927920799702953109, 5.69003541410506476139286667585, 6.72995060631504894236144495922, 7.47258036429880613369264215433, 8.815583880703080147166424819927, 9.255109555070225102644864004464, 10.32323225063286106676204367390
