L(s) = 1 | + 6.54·5-s + 0.687i·7-s − 12.1i·11-s − 8.29·13-s + 13.7·17-s + 4.35i·19-s + 42.5i·23-s + 17.8·25-s − 53.4·29-s + 36.9i·31-s + 4.49i·35-s + 24.7·37-s + 79.9·41-s − 4.56i·43-s + 78.0i·47-s + ⋯ |
L(s) = 1 | + 1.30·5-s + 0.0981i·7-s − 1.10i·11-s − 0.638·13-s + 0.811·17-s + 0.229i·19-s + 1.84i·23-s + 0.713·25-s − 1.84·29-s + 1.19i·31-s + 0.128i·35-s + 0.668·37-s + 1.95·41-s − 0.106i·43-s + 1.65i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.674146995\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.674146995\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 - 6.54T + 25T^{2} \) |
| 7 | \( 1 - 0.687iT - 49T^{2} \) |
| 11 | \( 1 + 12.1iT - 121T^{2} \) |
| 13 | \( 1 + 8.29T + 169T^{2} \) |
| 17 | \( 1 - 13.7T + 289T^{2} \) |
| 23 | \( 1 - 42.5iT - 529T^{2} \) |
| 29 | \( 1 + 53.4T + 841T^{2} \) |
| 31 | \( 1 - 36.9iT - 961T^{2} \) |
| 37 | \( 1 - 24.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 79.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + 4.56iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 78.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 13.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 43.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 112. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 8.14iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 4.57iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 107.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 98.4T + 9.40e3T^{2} \) |
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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
−9.001236753627708365090452879945, −7.80933884013158378708183319209, −7.33823339263219447026363874466, −6.05160888879031162511961754893, −5.78061937532769382288454551528, −5.09661969373563648433871196696, −3.79035208207266138666215033645, −2.95398052046308497610658440256, −1.95390543034345822906236576246, −1.00410622157216999863226767527,
0.68753873819545242601733972737, 2.12535139297494721356185204158, 2.40167656755429728437271533171, 3.88962599727634422705503102511, 4.74286061938298262053309152623, 5.57404120584780904669223731665, 6.17481850339186109363531747321, 7.12751393815486837562993874991, 7.65777974379092012072912147567, 8.748214152657157370005359436900