L(s) = 1 | + (−0.980 − 0.195i)3-s + (0.555 − 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 + 0.195i)21-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.831 − 0.555i)27-s + (−0.195 + 0.980i)29-s − i·31-s − i·33-s + ⋯ |
L(s) = 1 | + (−0.980 − 0.195i)3-s + (0.555 − 0.831i)5-s + (0.923 − 0.382i)7-s + (0.923 + 0.382i)9-s + (0.195 + 0.980i)11-s + (0.555 + 0.831i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.980 + 0.195i)21-s + (−0.382 + 0.923i)23-s + (−0.382 − 0.923i)25-s + (−0.831 − 0.555i)27-s + (−0.195 + 0.980i)29-s − i·31-s − i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.274902643 - 0.7222234264i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.274902643 - 0.7222234264i\) |
\(L(1)\) |
\(\approx\) |
\(0.9769636717 - 0.2179928867i\) |
\(L(1)\) |
\(\approx\) |
\(0.9769636717 - 0.2179928867i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.980 - 0.195i)T \) |
| 5 | \( 1 + (0.555 - 0.831i)T \) |
| 7 | \( 1 + (0.923 - 0.382i)T \) |
| 11 | \( 1 + (0.195 + 0.980i)T \) |
| 13 | \( 1 + (0.555 + 0.831i)T \) |
| 17 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 + (-0.382 + 0.923i)T \) |
| 29 | \( 1 + (-0.195 + 0.980i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.831 - 0.555i)T \) |
| 41 | \( 1 + (0.382 - 0.923i)T \) |
| 43 | \( 1 + (0.980 - 0.195i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (-0.195 - 0.980i)T \) |
| 59 | \( 1 + (-0.555 + 0.831i)T \) |
| 61 | \( 1 + (0.980 + 0.195i)T \) |
| 67 | \( 1 + (0.980 + 0.195i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 + (-0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.707 - 0.707i)T \) |
| 83 | \( 1 + (-0.831 + 0.555i)T \) |
| 89 | \( 1 + (-0.382 - 0.923i)T \) |
| 97 | \( 1 - iT \) |
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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
−19.44293469643665967940575510814, −18.60134979391088895694088001584, −18.19505363284433878766677950411, −17.43810425785040711551733247996, −17.107856512117934790615543946903, −15.93404759970065879228536764845, −15.48529465853047578088330980539, −14.61729134855549763616786930410, −13.98775805918873270832551291727, −13.09803536903458792769759662257, −12.35913410509878209252742922730, −11.343525051577509043647806820530, −10.99546921804876005710348111292, −10.46392528695478964451803917381, −9.58640390500645223137666124973, −8.55264103737940928133346487556, −7.92557402466082847958203504994, −6.76103415158334033176567696761, −6.13168645048045411738871697341, −5.66792702220434527467806261400, −4.76567513873155422770951628367, −3.86195046831391583321003552913, −2.89274279849985958077464596958, −1.872555474326135202570186659277, −0.949342122203456843672046355771,
0.65791548964974424515651594810, 1.794270257363169075170578470674, 1.94057866155834808953416932377, 3.93965125496004946674243958940, 4.49816781374834833375299359599, 5.19956400925433210533665198108, 5.82954790059630569987247767533, 6.91049792124714408539061106082, 7.33880177352315185711712790871, 8.42175194222195096073927462741, 9.26587867090650163159993576506, 9.90251803231890340949155414510, 10.85325302132888436648926127620, 11.469283304037746941543680819, 12.10633950605095062170161558989, 12.84903653758039359736545403795, 13.61740333614977336475835599110, 14.15614977277276511376024321998, 15.25438486363216490671992255408, 16.0613245957927950092977132689, 16.62131156160123294444016039763, 17.489222004936936671778543672339, 17.70584154638933355940676989552, 18.37435737933823794863080808378, 19.39704922140806065463586553657