L(s) = 1 | + (−1.45 + 1.45i)2-s − 1.81i·3-s − 2.21i·4-s + (−2.01 − 2.01i)5-s + (2.64 + 2.64i)6-s + (−1.13 − 2.38i)7-s + (0.311 + 0.311i)8-s − 0.311·9-s + 5.84·10-s + (0.451 + 0.451i)11-s − 4.02·12-s + (3.40 + 1.19i)13-s + (5.11 + 1.81i)14-s + (−3.66 + 3.66i)15-s + 3.52·16-s − 4.32·17-s + ⋯ |
L(s) = 1 | + (−1.02 + 1.02i)2-s − 1.05i·3-s − 1.10i·4-s + (−0.900 − 0.900i)5-s + (1.07 + 1.07i)6-s + (−0.429 − 0.903i)7-s + (0.109 + 0.109i)8-s − 0.103·9-s + 1.84·10-s + (0.136 + 0.136i)11-s − 1.16·12-s + (0.943 + 0.330i)13-s + (1.36 + 0.486i)14-s + (−0.946 + 0.946i)15-s + 0.881·16-s − 1.04·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.382950 - 0.253704i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.382950 - 0.253704i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (1.13 + 2.38i)T \) |
| 13 | \( 1 + (-3.40 - 1.19i)T \) |
good | 2 | \( 1 + (1.45 - 1.45i)T - 2iT^{2} \) |
| 3 | \( 1 + 1.81iT - 3T^{2} \) |
| 5 | \( 1 + (2.01 + 2.01i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.451 - 0.451i)T + 11iT^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + (3.40 + 3.40i)T + 19iT^{2} \) |
| 23 | \( 1 - 0.933iT - 23T^{2} \) |
| 29 | \( 1 - 6.33T + 29T^{2} \) |
| 31 | \( 1 + (-5.47 - 5.47i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.14 - 2.14i)T + 37iT^{2} \) |
| 41 | \( 1 + (-1.81 - 1.81i)T + 41iT^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (-5.90 + 5.90i)T - 47iT^{2} \) |
| 53 | \( 1 - 3.36T + 53T^{2} \) |
| 59 | \( 1 + (0.255 - 0.255i)T - 59iT^{2} \) |
| 61 | \( 1 - 7.78iT - 61T^{2} \) |
| 67 | \( 1 + (-7.28 + 7.28i)T - 67iT^{2} \) |
| 71 | \( 1 + (-5.56 + 5.56i)T - 71iT^{2} \) |
| 73 | \( 1 + (8.86 - 8.86i)T - 73iT^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + (-4.30 - 4.30i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.61 + 5.61i)T - 89iT^{2} \) |
| 97 | \( 1 + (0.236 + 0.236i)T + 97iT^{2} \) |
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
−13.77486074439034149983297155167, −12.93622941433560973143171754846, −11.92386257804758967180702034176, −10.42055631421656241055246977426, −8.876086172390961923142255764921, −8.242563186318682345480721552789, −7.08188394131224620869164390284, −6.51077263771438043937067101725, −4.27092141553524445919487173591, −0.816151103817081694719927913812,
2.81146420929664847500361139621, 4.04595004440018051472213196042, 6.23939874224492284056928273416, 8.129723040987991036688666960585, 9.057718779604643557384275476127, 10.12104077242343463341119190727, 10.90773872102485708980811476687, 11.59610215321874776925129784036, 12.79260158569794243276418745787, 14.66842980712798614978897452643