L(s) = 1 | − 2i·5-s + (2.23 − 1.41i)7-s − 1.41·11-s + 4.47·17-s + 6.32·19-s − 3.16i·23-s + 25-s − 3.16·29-s + 5.65i·31-s + (−2.82 − 4.47i)35-s − 4.47i·37-s + 4.47·41-s + 4i·43-s + 8·47-s + (3.00 − 6.32i)49-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + (0.845 − 0.534i)7-s − 0.426·11-s + 1.08·17-s + 1.45·19-s − 0.659i·23-s + 0.200·25-s − 0.587·29-s + 1.01i·31-s + (−0.478 − 0.755i)35-s − 0.735i·37-s + 0.698·41-s + 0.609i·43-s + 1.16·47-s + (0.428 − 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.259505124\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.259505124\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 + 3.16iT - 23T^{2} \) |
| 29 | \( 1 + 3.16T + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 4.47iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 9.48iT - 71T^{2} \) |
| 73 | \( 1 - 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.32iT - 97T^{2} \) |
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
−8.245892476968189091518968699330, −7.62527755408726812260824318547, −7.10289530367415521714471202957, −5.84950751936036500188785614393, −5.22923984808884574072963746116, −4.67580478282357054250349901537, −3.77526365379041677770651815138, −2.79767189411040286462902307975, −1.50599181665335712718699869826, −0.78535435817670845648564722556,
1.12767580719912969012009912035, 2.27695709997569167446503436784, 3.06416911732364035769728536637, 3.85448320516047175546696792187, 5.07905074334282927462465850701, 5.50358649256593844225482155777, 6.32124387928290515137987325875, 7.41858596385834477728522048105, 7.63663119046530136787542776158, 8.447414206274547472498593800019