L(s) = 1 | + (13.1 + 54.3i)5-s − 146. i·7-s + 191.·11-s − 83.9i·13-s + 2.00e3i·17-s + 677.·19-s − 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s − 3.26e3·29-s − 6.15e3·31-s + (7.97e3 − 1.93e3i)35-s − 1.13e4i·37-s + 1.05e4·41-s + 1.29e4i·43-s + 9.52e3i·47-s + ⋯ |
L(s) = 1 | + (0.235 + 0.971i)5-s − 1.13i·7-s + 0.476·11-s − 0.137i·13-s + 1.67i·17-s + 0.430·19-s − 0.510i·23-s + (−0.889 + 0.457i)25-s − 0.721·29-s − 1.15·31-s + (1.10 − 0.266i)35-s − 1.36i·37-s + 0.984·41-s + 1.06i·43-s + 0.628i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.646412068\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.646412068\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-13.1 - 54.3i)T \) |
good | 7 | \( 1 + 146. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 191.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 83.9iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.00e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 677.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.29e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.26e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.13e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.05e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 9.52e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 1.47e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.82e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.58e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.17e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.33e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.59e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.33e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.08e4iT - 8.58e9T^{2} \) |
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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
−10.01107623221740619884121549118, −9.189177067266470430787973864274, −7.952279362495052154422455909593, −7.27925718178122363516563610940, −6.44436110684618593924350566270, −5.63443166967463630863525127279, −4.10335060159638346953169000086, −3.59318526703577073335998981585, −2.23418420605270386679722770888, −1.08711662988951257461182819972,
0.36171891910173093535228290110, 1.58135566858768221600228768031, 2.61658104911642400989684470063, 3.89873082930688006727976039435, 5.18530626364323829444221298153, 5.49956990078726753464623906378, 6.74694169187807298907596136619, 7.74967700142122574007179509615, 8.829338826891346711026325831924, 9.234798881905663620289802308134