L(s) = 1 | + (−0.407 + 0.407i)2-s + (−0.544 + 0.544i)3-s + 1.66i·4-s + (0.752 + 2.10i)5-s − 0.443i·6-s + (0.843 − 0.843i)7-s + (−1.49 − 1.49i)8-s + 2.40i·9-s + (−1.16 − 0.550i)10-s + (−0.908 − 0.908i)12-s + (4.29 + 4.29i)13-s + 0.687i·14-s + (−1.55 − 0.736i)15-s − 2.12·16-s + (−0.262 + 0.262i)17-s + (−0.980 − 0.980i)18-s + ⋯ |
L(s) = 1 | + (−0.287 + 0.287i)2-s + (−0.314 + 0.314i)3-s + 0.834i·4-s + (0.336 + 0.941i)5-s − 0.180i·6-s + (0.318 − 0.318i)7-s + (−0.528 − 0.528i)8-s + 0.802i·9-s + (−0.368 − 0.174i)10-s + (−0.262 − 0.262i)12-s + (1.19 + 1.19i)13-s + 0.183i·14-s + (−0.401 − 0.190i)15-s − 0.530·16-s + (−0.0635 + 0.0635i)17-s + (−0.231 − 0.231i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.255196 + 1.10736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.255196 + 1.10736i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.752 - 2.10i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.407 - 0.407i)T - 2iT^{2} \) |
| 3 | \( 1 + (0.544 - 0.544i)T - 3iT^{2} \) |
| 7 | \( 1 + (-0.843 + 0.843i)T - 7iT^{2} \) |
| 13 | \( 1 + (-4.29 - 4.29i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.262 - 0.262i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.37T + 19T^{2} \) |
| 23 | \( 1 + (-3.48 + 3.48i)T - 23iT^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + (-2.10 - 2.10i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.12iT - 41T^{2} \) |
| 43 | \( 1 + (6.75 + 6.75i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.902 - 0.902i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.288 - 0.288i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.187iT - 59T^{2} \) |
| 61 | \( 1 + 0.683iT - 61T^{2} \) |
| 67 | \( 1 + (-7.14 - 7.14i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.03T + 71T^{2} \) |
| 73 | \( 1 + (-3.23 - 3.23i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-8.04 - 8.04i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.04iT - 89T^{2} \) |
| 97 | \( 1 + (-5.96 - 5.96i)T + 97iT^{2} \) |
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
−11.21628029428149319581966216678, −10.22925544128103942049369635862, −9.200516911755262027084232377260, −8.382497457524560676768329520770, −7.36136354768479784418890928621, −6.76632272731776755804535543246, −5.71406629993402622105427678068, −4.36395223061290789966926214547, −3.45720045551383141667712735597, −2.07464475409174363980822368304,
0.75898732446920784276342317012, 1.69566692694539111757777843397, 3.42305687713360379640567037040, 5.01545577578707181184722206245, 5.69139151068043982308245967476, 6.35887350577027259289166515658, 7.79654770243426025626341931929, 8.811939791012647446691869242849, 9.344375087212404714341929064908, 10.22108930316680330174301119065