L(s) = 1 | − 2.85·5-s + (2.30 − 1.29i)7-s − 3.55i·11-s − 2.43i·13-s + 0.860·17-s + 3.55i·19-s − 5.94i·23-s + 3.15·25-s − 2.88i·29-s + 9.01i·31-s + (−6.58 + 3.70i)35-s + 7.43·37-s − 6.85·41-s − 3.48·43-s − 0.263·47-s + ⋯ |
L(s) = 1 | − 1.27·5-s + (0.871 − 0.489i)7-s − 1.07i·11-s − 0.675i·13-s + 0.208·17-s + 0.816i·19-s − 1.23i·23-s + 0.631·25-s − 0.535i·29-s + 1.61i·31-s + (−1.11 + 0.625i)35-s + 1.22·37-s − 1.07·41-s − 0.532·43-s − 0.0384·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7725576733\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7725576733\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.30 + 1.29i)T \) |
good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 3.55iT - 11T^{2} \) |
| 13 | \( 1 + 2.43iT - 13T^{2} \) |
| 17 | \( 1 - 0.860T + 17T^{2} \) |
| 19 | \( 1 - 3.55iT - 19T^{2} \) |
| 23 | \( 1 + 5.94iT - 23T^{2} \) |
| 29 | \( 1 + 2.88iT - 29T^{2} \) |
| 31 | \( 1 - 9.01iT - 31T^{2} \) |
| 37 | \( 1 - 7.43T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 + 3.48T + 43T^{2} \) |
| 47 | \( 1 + 0.263T + 47T^{2} \) |
| 53 | \( 1 + 7.76iT - 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 - 1.62iT - 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 + 9.20iT - 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 0.922T + 83T^{2} \) |
| 89 | \( 1 - 4.90T + 89T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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
−8.170772839733194170122054265498, −7.951230938543782061039624711721, −7.05333376730630897222999630522, −6.15799835016965956280435583522, −5.20435529976040831216737486027, −4.42532756766906986049857251452, −3.66268243766790417020687880897, −2.91397596105138745819090521083, −1.36756641824228951354676192625, −0.26082198968565517885918387187,
1.43372201411854651245348145693, 2.46485156927664489150835680568, 3.62100341766712604190865273965, 4.46397596824156485774456476069, 4.91849414803177812266262265216, 5.99695410937653094758699616313, 7.02195767847487670155268859915, 7.68207403968966488269281764308, 8.026306658688862188244855806434, 9.121278150063967814116562079651