| L(s) = 1 | + (−0.111 − 1.40i)2-s + (0.583 + 0.266i)3-s + (−1.97 + 0.313i)4-s + (0.251 − 0.856i)5-s + (0.310 − 0.851i)6-s + (0.423 − 1.22i)7-s + (0.661 + 2.75i)8-s + (−1.69 − 1.95i)9-s + (−1.23 − 0.259i)10-s + (4.42 − 1.07i)11-s + (−1.23 − 0.343i)12-s + (−0.306 − 0.594i)13-s + (−1.77 − 0.461i)14-s + (0.374 − 0.432i)15-s + (3.80 − 1.23i)16-s + (−3.86 − 1.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.0785 − 0.996i)2-s + (0.336 + 0.153i)3-s + (−0.987 + 0.156i)4-s + (0.112 − 0.383i)5-s + (0.126 − 0.347i)6-s + (0.160 − 0.462i)7-s + (0.233 + 0.972i)8-s + (−0.565 − 0.652i)9-s + (−0.390 − 0.0820i)10-s + (1.33 − 0.323i)11-s + (−0.356 − 0.0991i)12-s + (−0.0850 − 0.164i)13-s + (−0.473 − 0.123i)14-s + (0.0967 − 0.111i)15-s + (0.950 − 0.309i)16-s + (−0.936 − 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.451668 - 1.20635i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.451668 - 1.20635i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.111 + 1.40i)T \) |
| 67 | \( 1 + (-2.67 + 7.73i)T \) |
| good | 3 | \( 1 + (-0.583 - 0.266i)T + (1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-0.251 + 0.856i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.423 + 1.22i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-4.42 + 1.07i)T + (9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (0.306 + 0.594i)T + (-7.54 + 10.5i)T^{2} \) |
| 17 | \( 1 + (3.86 + 1.54i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (0.115 - 0.0398i)T + (14.9 - 11.7i)T^{2} \) |
| 23 | \( 1 + (3.63 + 5.09i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (-1.57 + 0.909i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (7.23 + 3.73i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (2.12 + 1.22i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.04 + 3.96i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (-0.0424 + 0.00610i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + (4.62 - 0.441i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (-9.46 - 1.36i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-2.56 - 3.99i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.06 - 0.500i)T + (54.2 + 27.9i)T^{2} \) |
| 71 | \( 1 + (-3.07 + 1.23i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (3.15 - 13.0i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.251 - 5.28i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-9.68 - 10.1i)T + (-3.94 + 82.9i)T^{2} \) |
| 89 | \( 1 + (-3.56 - 7.81i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-0.545 + 0.944i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.60103759548203821475311675731, −9.437994767106845642515110303068, −9.007493302922703981522059322114, −8.251557666386460461279112209094, −6.86626057594150838944668238289, −5.66506375281546176395814170327, −4.34099228000281560721769394439, −3.68359241622018527507877217305, −2.34267416333220521890907038611, −0.77701258667674670390409968582,
1.91527169468017610413322612736, 3.57223259381010443631721182368, 4.74455860250258881902591910953, 5.81672756937570328904351666860, 6.69354678707632989839773033403, 7.48690064774078389098852116744, 8.605065216899001641867757659241, 9.004685817407712137052472640695, 10.05740713180858306169297343581, 11.12615672638193785002509063217
