L(s) = 1 | + (−2.44 − 4.58i)3-s − 2.31·5-s + 29.6i·7-s + (−15 + 22.4i)9-s − 22.8i·11-s − 8.66i·13-s + (5.66 + 10.6i)15-s − 93.3i·17-s + 85.4·19-s + (135. − 72.6i)21-s + 116.·23-s − 119.·25-s + (139. + 13.7i)27-s − 103.·29-s − 10.6i·31-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.881i)3-s − 0.207·5-s + 1.60i·7-s + (−0.555 + 0.831i)9-s − 0.625i·11-s − 0.184i·13-s + (0.0975 + 0.182i)15-s − 1.33i·17-s + 1.03·19-s + (1.41 − 0.755i)21-s + 1.05·23-s − 0.957·25-s + (0.995 + 0.0979i)27-s − 0.662·29-s − 0.0614i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.045719097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.045719097\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.44 + 4.58i)T \) |
good | 5 | \( 1 + 2.31T + 125T^{2} \) |
| 7 | \( 1 - 29.6iT - 343T^{2} \) |
| 11 | \( 1 + 22.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 8.66iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 93.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 10.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 380. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 257. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 359.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 679.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 45.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 595. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 593.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 910. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 332. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 821. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 420.T + 9.12e5T^{2} \) |
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
−11.05495045417006206688760147303, −9.532363971773938324467047827046, −8.754916080252737357594828354750, −7.77421222434977744817878065656, −6.85785440811193895837761307691, −5.62194567528563450031084560443, −5.25321819802951563985873285120, −3.17641306744327617180398276441, −2.09187824509514195469510534669, −0.42779299956063003281643348859,
1.18551483360187471304988402036, 3.42543556098329336507812365572, 4.19486153067814323169476375310, 5.13514648531272822878453173201, 6.46609948058308520088963008769, 7.35356791090999351712919537949, 8.433065224697915467312746124820, 9.799660669341962611255292407415, 10.13961949936861558525427209184, 11.12490987203404949015041961641