| L(s) = 1 | + (−1.41 + 1.73i)5-s − 2.82i·7-s + 2i·11-s − 2.82·13-s − 4.89i·17-s + 5.65i·23-s + (−0.999 − 4.89i)25-s + 3.46i·29-s + 3.46·31-s + (4.89 + 4.00i)35-s − 2.82·37-s + 8·41-s + 9.79·43-s − 1.00·49-s + 8.48·53-s + ⋯ |
| L(s) = 1 | + (−0.632 + 0.774i)5-s − 1.06i·7-s + 0.603i·11-s − 0.784·13-s − 1.18i·17-s + 1.17i·23-s + (−0.199 − 0.979i)25-s + 0.643i·29-s + 0.622·31-s + (0.828 + 0.676i)35-s − 0.464·37-s + 1.24·41-s + 1.49·43-s − 0.142·49-s + 1.16·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336186570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336186570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.41 - 1.73i)T \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.927523111809718607836421322509, −7.63949717779665880225348837692, −7.41426799381643414889162701567, −6.92586868444661612832857277573, −5.82604739741060556337986247727, −4.76194872153295836529503583762, −4.13815523969791746070076120018, −3.22507764425853851392505694071, −2.35332024311452558763156312396, −0.831747298220392909803647720910,
0.59360037402947904576392282241, 2.03102367376427619081996467997, 2.95436043747532053858895605813, 4.09254743307312177857402621690, 4.72401778501453818317440231375, 5.72218995213575347065136613519, 6.17359852422529055379660714722, 7.38164864019686447094267438290, 8.064413379176309709666120484430, 8.766413668420071853115000758582
