L(s) = 1 | − 5.02i·2-s − 17.2·4-s + (−5.75 − 9.58i)5-s + 5.38i·7-s + 46.4i·8-s + (−48.1 + 28.9i)10-s + 39.1·11-s + 86.6i·13-s + 27.0·14-s + 95.2·16-s + 15.4i·17-s + 26.8·19-s + (99.2 + 165. i)20-s − 196. i·22-s + 111. i·23-s + ⋯ |
L(s) = 1 | − 1.77i·2-s − 2.15·4-s + (−0.514 − 0.857i)5-s + 0.290i·7-s + 2.05i·8-s + (−1.52 + 0.914i)10-s + 1.07·11-s + 1.84i·13-s + 0.516·14-s + 1.48·16-s + 0.219i·17-s + 0.324·19-s + (1.10 + 1.84i)20-s − 1.90i·22-s + 1.00i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.161589950\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161589950\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (5.75 + 9.58i)T \) |
good | 2 | \( 1 + 5.02iT - 8T^{2} \) |
| 7 | \( 1 - 5.38iT - 343T^{2} \) |
| 11 | \( 1 - 39.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 15.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 26.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 111. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 49.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 293. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 27.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 60.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 96.2iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 251. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 76.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 238. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 640.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 769. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 662.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 995.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 814. iT - 9.12e5T^{2} \) |
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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
−11.09794706709702589209293875136, −9.608116830235560043685184362772, −9.242803177187154825368237030730, −8.497911045829955523824530940984, −7.02196828843592411674261428065, −5.40521999153573473520281829995, −4.20617967356510879717940289533, −3.73730397971933669875642411182, −2.04315262497542045900212827515, −1.13626210309839831616397403449,
0.47242124316574013651271313146, 3.20374319279446846149865576524, 4.32021433902287165304380117657, 5.51185993216871953510002495344, 6.44618701993218952358740440136, 7.18187855556011798520411752567, 7.939970204488147029912100141351, 8.726655849799298858885342472528, 9.903978084715038132696600458788, 10.76519078004407246854686299755