L(s) = 1 | + (0.331 + 1.37i)2-s + 2.35·3-s + (−1.78 + 0.910i)4-s − 2.56·5-s + (0.780 + 3.24i)6-s − 4.15i·7-s + (−1.84 − 2.14i)8-s + 2.56·9-s + (−0.848 − 3.52i)10-s + 2.33i·11-s + (−4.19 + 2.14i)12-s + 4.29i·13-s + (5.70 − 1.37i)14-s − 6.04·15-s + (2.34 − 3.24i)16-s − 17-s + ⋯ |
L(s) = 1 | + (0.234 + 0.972i)2-s + 1.36·3-s + (−0.890 + 0.455i)4-s − 1.14·5-s + (0.318 + 1.32i)6-s − 1.56i·7-s + (−0.650 − 0.759i)8-s + 0.853·9-s + (−0.268 − 1.11i)10-s + 0.703i·11-s + (−1.21 + 0.619i)12-s + 1.19i·13-s + (1.52 − 0.367i)14-s − 1.55·15-s + (0.585 − 0.810i)16-s − 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.02577 + 0.585356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.02577 + 0.585356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.331 - 1.37i)T \) |
| 19 | \( 1 + (-3.68 + 2.33i)T \) |
good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 + 4.15iT - 7T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 - 4.29iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 - 1.82iT - 23T^{2} \) |
| 29 | \( 1 - 1.20iT - 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 - 5.49iT - 41T^{2} \) |
| 43 | \( 1 + 1.30iT - 43T^{2} \) |
| 47 | \( 1 + 6.99iT - 47T^{2} \) |
| 53 | \( 1 - 9.79iT - 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 0.290T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 0.123T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 + 14.0iT - 89T^{2} \) |
| 97 | \( 1 + 2.41iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.60006337977813350569178969393, −13.93133851744439537543564036865, −13.09101454595466601019012905474, −11.57505136425184562866212083122, −9.791228390138686301428530805517, −8.703488945975527412155544133601, −7.47962663616793606517488996346, −7.15159679734493210836232362493, −4.42148515038659471959616264176, −3.64174237733015602439561412964,
2.67371941033333324457382277444, 3.62028145778217057486891019273, 5.49894328770866825014555193549, 8.099422338167298163797584798119, 8.564326940179218256593446838710, 9.726777410095055001861729912255, 11.29093668359761925907665799131, 12.18144283195207747099499605380, 13.14041219027240167657850564905, 14.35805513301153449838420219782