L(s) = 1 | + (2.41 − 2.41i)5-s − 1.53i·7-s + (3.37 − 3.37i)11-s + (−0.414 − 0.414i)13-s − 2.82·17-s + (0.317 + 0.317i)19-s − 5.86i·23-s − 6.65i·25-s + (−3.24 − 3.24i)29-s − 7.39·31-s + (−3.69 − 3.69i)35-s + (3.58 − 3.58i)37-s + 4i·41-s + (−1.84 + 1.84i)43-s − 7.39·47-s + ⋯ |
L(s) = 1 | + (1.07 − 1.07i)5-s − 0.578i·7-s + (1.01 − 1.01i)11-s + (−0.114 − 0.114i)13-s − 0.685·17-s + (0.0727 + 0.0727i)19-s − 1.22i·23-s − 1.33i·25-s + (−0.602 − 0.602i)29-s − 1.32·31-s + (−0.624 − 0.624i)35-s + (0.589 − 0.589i)37-s + 0.624i·41-s + (−0.281 + 0.281i)43-s − 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.084058238\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.084058238\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.41 + 2.41i)T - 5iT^{2} \) |
| 7 | \( 1 + 1.53iT - 7T^{2} \) |
| 11 | \( 1 + (-3.37 + 3.37i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.414 + 0.414i)T + 13iT^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-0.317 - 0.317i)T + 19iT^{2} \) |
| 23 | \( 1 + 5.86iT - 23T^{2} \) |
| 29 | \( 1 + (3.24 + 3.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + (-3.58 + 3.58i)T - 37iT^{2} \) |
| 41 | \( 1 - 4iT - 41T^{2} \) |
| 43 | \( 1 + (1.84 - 1.84i)T - 43iT^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + (5.24 - 5.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.84 + 1.84i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.24 - 9.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (7.07 + 7.07i)T + 67iT^{2} \) |
| 71 | \( 1 + 11.9iT - 71T^{2} \) |
| 73 | \( 1 - 10.4iT - 73T^{2} \) |
| 79 | \( 1 - 6.12T + 79T^{2} \) |
| 83 | \( 1 + (-2.48 - 2.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.828iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.235539730818148961542064969444, −7.30756883354144035550004344083, −6.34165193119387168058702256686, −5.98612621138381141578649470450, −5.08514139021306583495520215863, −4.35740021872774184338001790055, −3.58630434061054085061895994858, −2.36113298531816451154477593836, −1.43709692688999567739477180668, −0.54063107941873493671479739961,
1.69905062935868416396924989287, 2.10405051288651657384971029825, 3.15851811930828321152686284016, 3.97016116717315529250795474472, 5.08466765559603804156710951399, 5.70823740106185254324628375391, 6.56621753969940048806758618742, 6.93012352123695659932498373152, 7.66919808743913834659417920329, 8.853883258090056620556514126363