L(s) = 1 | + (−7.48 − 7.48i)3-s + (−19.2 + 19.2i)7-s − 131. i·9-s − 180. i·11-s + (−44.2 + 44.2i)13-s + (621. + 621. i)17-s + 2.67e3·19-s + 287.·21-s + (−2.23e3 − 2.23e3i)23-s + (−2.79e3 + 2.79e3i)27-s − 705. i·29-s − 2.76e3i·31-s + (−1.35e3 + 1.35e3i)33-s + (3.54e3 + 3.54e3i)37-s + 661.·39-s + ⋯ |
L(s) = 1 | + (−0.480 − 0.480i)3-s + (−0.148 + 0.148i)7-s − 0.539i·9-s − 0.450i·11-s + (−0.0725 + 0.0725i)13-s + (0.521 + 0.521i)17-s + 1.69·19-s + 0.142·21-s + (−0.880 − 0.880i)23-s + (−0.738 + 0.738i)27-s − 0.155i·29-s − 0.516i·31-s + (−0.216 + 0.216i)33-s + (0.425 + 0.425i)37-s + 0.0696·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.011398281\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011398281\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (7.48 + 7.48i)T + 243iT^{2} \) |
| 7 | \( 1 + (19.2 - 19.2i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + 180. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (44.2 - 44.2i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (-621. - 621. i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.23e3 + 2.23e3i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 705. iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.76e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-3.54e3 - 3.54e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.09e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + (5.34e3 + 5.34e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (-1.32e4 + 1.32e4i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-1.56e4 + 1.56e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 4.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.75e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-3.10e4 + 3.10e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 1.06e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (5.09e4 - 5.09e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 2.47e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (4.89e4 + 4.89e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 2.61e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (3.99e4 + 3.99e4i)T + 8.58e9iT^{2} \) |
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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
−10.04124444654467692656893639903, −9.240528242403332183067535349612, −8.120978540951432184813211593865, −7.20329068732234389829204412890, −6.15998508807161369010466532263, −5.52905893637239161903358073302, −4.05498370307076342701916566544, −2.90887951688020808865324002224, −1.35595837749519278103062343107, −0.29424390781487581980526424777,
1.25737456047239116134134240870, 2.78401035387803481303647170285, 4.05479769733606901618161779220, 5.13179362841029388652891381060, 5.83117496482327899179235543575, 7.27611414799444039223219269905, 7.85996110892331572630314053495, 9.344569284933051143793004713559, 9.912000166913618869777867104510, 10.80886795930712482151281598154