L(s) = 1 | + (2.74 + 4.41i)3-s − 14.3·5-s + 18.8·7-s + (−11.9 + 24.2i)9-s − 26.9i·11-s + (45.5 + 11.0i)13-s + (−39.2 − 63.1i)15-s + 26.8i·17-s + 108.·19-s + (51.7 + 83.3i)21-s + 14.6·23-s + 79.6·25-s + (−139. + 13.6i)27-s + 249. i·29-s + 29.9·31-s + ⋯ |
L(s) = 1 | + (0.527 + 0.849i)3-s − 1.27·5-s + 1.01·7-s + (−0.442 + 0.896i)9-s − 0.739i·11-s + (0.971 + 0.235i)13-s + (−0.675 − 1.08i)15-s + 0.383i·17-s + 1.30·19-s + (0.538 + 0.865i)21-s + 0.132·23-s + 0.636·25-s + (−0.995 + 0.0975i)27-s + 1.59i·29-s + 0.173·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.936785657\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.936785657\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.74 - 4.41i)T \) |
| 13 | \( 1 + (-45.5 - 11.0i)T \) |
good | 5 | \( 1 + 14.3T + 125T^{2} \) |
| 7 | \( 1 - 18.8T + 343T^{2} \) |
| 11 | \( 1 + 26.9iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 26.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 14.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 249. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 29.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 177.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 319. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 394. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 130. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 38.7iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 25.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 56.9T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.03e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 760. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.01e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 300.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.29e3iT - 9.12e5T^{2} \) |
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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
−10.76128542154017985113160861709, −9.458946024171463835305518010583, −8.519650169153804339905725396540, −8.127885537476083401227854697983, −7.23211411704335145587399724800, −5.68356037974025205586809858757, −4.72042246311266793270602800364, −3.82609808546449531808289303299, −3.08871065296493336216313355226, −1.27768807193348317812528712135,
0.59099503578053225949217778916, 1.78421533528202334528178678246, 3.19346436041413942098093989680, 4.14538771654084252532574507450, 5.29107050678569012659474570942, 6.64371546739873035216796056525, 7.57339081993680998504270766168, 7.996065601328193939705826761719, 8.734643211996294791687647527571, 9.860059490654456830535454933733