L(s) = 1 | + 0.874·3-s − 3.70i·5-s + (−1.41 − 2.23i)7-s − 2.23·9-s + 3.23i·11-s + 0.874i·13-s − 3.23i·15-s − 4.57i·17-s − 1.95·19-s + (−1.23 − 1.95i)21-s − 1.23i·23-s − 8.70·25-s − 4.57·27-s − 2·29-s + 10.2·31-s + ⋯ |
L(s) = 1 | + 0.504·3-s − 1.65i·5-s + (−0.534 − 0.845i)7-s − 0.745·9-s + 0.975i·11-s + 0.242i·13-s − 0.835i·15-s − 1.10i·17-s − 0.448·19-s + (−0.269 − 0.426i)21-s − 0.257i·23-s − 1.74·25-s − 0.880·27-s − 0.371·29-s + 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.305400 - 1.05423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305400 - 1.05423i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.41 + 2.23i)T \) |
good | 3 | \( 1 - 0.874T + 3T^{2} \) |
| 5 | \( 1 + 3.70iT - 5T^{2} \) |
| 11 | \( 1 - 3.23iT - 11T^{2} \) |
| 13 | \( 1 - 0.874iT - 13T^{2} \) |
| 17 | \( 1 + 4.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 1.23iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 3.90iT - 41T^{2} \) |
| 43 | \( 1 - 1.70iT - 43T^{2} \) |
| 47 | \( 1 + 8.07T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 8.94iT - 61T^{2} \) |
| 67 | \( 1 - 9.70iT - 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + 14.1iT - 73T^{2} \) |
| 79 | \( 1 + 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 9.82iT - 89T^{2} \) |
| 97 | \( 1 - 4.57iT - 97T^{2} \) |
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
−9.588093383301745468038758600711, −8.958429788299529436309075805730, −8.239686129346459957581591073653, −7.36861625374541787083698736802, −6.36029762837596688212028724828, −5.07064852660337654481152238513, −4.51888826610682980342043049708, −3.37962029258533855201715818439, −1.96340014655408535178794482379, −0.46179733147297661444400625252,
2.27936874957916430277837892613, 3.06948334576679811002074066353, 3.67090105735052193011594692492, 5.54132539927007580249599892726, 6.22143090807189607469728668920, 6.88024615624690102734192431615, 8.222482630005076160882850008678, 8.529765355557943673842848838002, 9.733926598788514163756679091552, 10.43580426302653206016540190492