L(s) = 1 | + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s + (0.984 − 0.173i)22-s + (0.642 − 0.766i)23-s + (0.5 − 0.866i)26-s + (−0.342 − 0.939i)28-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (0.984 + 0.173i)32-s + ⋯ |
L(s) = 1 | + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (0.866 + 0.5i)7-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)11-s + (0.984 + 0.173i)13-s + (0.766 − 0.642i)14-s + (0.173 + 0.984i)16-s + (0.342 − 0.939i)17-s + (0.984 − 0.173i)22-s + (0.642 − 0.766i)23-s + (0.5 − 0.866i)26-s + (−0.342 − 0.939i)28-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (0.984 + 0.173i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.317983621 - 0.8017100778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.317983621 - 0.8017100778i\) |
\(L(1)\) |
\(\approx\) |
\(1.185696339 - 0.5462174706i\) |
\(L(1)\) |
\(\approx\) |
\(1.185696339 - 0.5462174706i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.984 + 0.173i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 23 | \( 1 + (0.642 - 0.766i)T \) |
| 29 | \( 1 + (-0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.342 - 0.939i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (-0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (-0.984 + 0.173i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (0.173 - 0.984i)T \) |
| 97 | \( 1 + (-0.342 + 0.939i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.67347620604243934082414242728, −24.67462032759207626123571566869, −23.90407981149877378519575411743, −23.30789558921764041712297899952, −22.19761477629026587444421076352, −21.342109684899987557383800028312, −20.54667886721943199277485980418, −19.1150798456995073470587481498, −18.236716237451702789056837984884, −17.1564071976647212080688790545, −16.68800780517919001627753996576, −15.45609798129813724610517141272, −14.68458326953783863747117683904, −13.73929400303445693825203649080, −13.07812762400786723934864708661, −11.68149910077171116770842352226, −10.80182892607150757225160218147, −9.295776830074078977775884174930, −8.3160085562207779629945618575, −7.5930425618009035310443987547, −6.305915767216260674061641218645, −5.50663458960972736269859851640, −4.215186035227039931621278235302, −3.39118862017991723742071364049, −1.275644252473797689562214883401,
1.31023245026887796112669269414, 2.34203968613801349467081231538, 3.70078046830134350590537767238, 4.77175872316265333141949042260, 5.66576571242482281585118683847, 7.12336720143494159886426243933, 8.621424379061662823981632671800, 9.29695020088018848918716789893, 10.560329475409516544399034642517, 11.41060689011822737746743663811, 12.19038951330358266428413843506, 13.14949544092929845489122991987, 14.32612324746602218578155856501, 14.83195035356380655002130351095, 16.11293092821545220051299190880, 17.54978749836217228623230568358, 18.24511590826014028602366698224, 19.01967816201793297821960921097, 20.275186257541542490092798594424, 20.78133473042944154256299570245, 21.64455844102557353277556197089, 22.67278144088390441131188955628, 23.30673603674770398511773740200, 24.40575555965016276971121601065, 25.2526534813094095667092760906