L(s) = 1 | + (−1.86 − 0.715i)2-s − 1.73i·3-s + (2.97 + 2.67i)4-s + 4.65i·5-s + (−1.23 + 3.23i)6-s − 2.30·7-s + (−3.64 − 7.12i)8-s − 2.99·9-s + (3.33 − 8.70i)10-s − 18.8·11-s + (4.62 − 5.15i)12-s + (−5.93 + 11.5i)13-s + (4.29 + 1.64i)14-s + 8.06·15-s + (1.71 + 15.9i)16-s − 7.23·17-s + ⋯ |
L(s) = 1 | + (−0.933 − 0.357i)2-s − 0.577i·3-s + (0.744 + 0.668i)4-s + 0.931i·5-s + (−0.206 + 0.539i)6-s − 0.328·7-s + (−0.455 − 0.890i)8-s − 0.333·9-s + (0.333 − 0.870i)10-s − 1.71·11-s + (0.385 − 0.429i)12-s + (−0.456 + 0.889i)13-s + (0.307 + 0.117i)14-s + 0.537·15-s + (0.107 + 0.994i)16-s − 0.425·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.210326 + 0.305457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.210326 + 0.305457i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.86 + 0.715i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 13 | \( 1 + (5.93 - 11.5i)T \) |
good | 5 | \( 1 - 4.65iT - 25T^{2} \) |
| 7 | \( 1 + 2.30T + 49T^{2} \) |
| 11 | \( 1 + 18.8T + 121T^{2} \) |
| 17 | \( 1 + 7.23T + 289T^{2} \) |
| 19 | \( 1 - 17.0T + 361T^{2} \) |
| 23 | \( 1 - 43.7iT - 529T^{2} \) |
| 29 | \( 1 + 52.6T + 841T^{2} \) |
| 31 | \( 1 + 10.5T + 961T^{2} \) |
| 37 | \( 1 - 12.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 38.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 62.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 74.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 11.4T + 3.48e3T^{2} \) |
| 61 | \( 1 + 32.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 60.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 8.07T + 5.04e3T^{2} \) |
| 73 | \( 1 - 58.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 59.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 48.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 80.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 15.3iT - 9.40e3T^{2} \) |
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
−12.92377527520832338806920433880, −11.73238472610372823932058352571, −10.98407068123482162399074220365, −10.01498486400040118338681273752, −9.014957612523874950151878342061, −7.43500214016624798528519143303, −7.28283453720987486452998840251, −5.69974240630706010601850638581, −3.30540058408076014034453677258, −2.13180881307856895901111680461,
0.29057252386868343888582144967, 2.68342393922974453122356381656, 4.90847106204568499425146324311, 5.73733887495147889087048537833, 7.43150726627162034157865004365, 8.338878874012426020954916014862, 9.282955913578118706968221709489, 10.24333958955545811690983347189, 10.95369641034327161303255198462, 12.40602971478671530098767810302