| L(s) = 1 | + (−2.18 + 0.456i)5-s − 0.913i·7-s + 3.58i·11-s + 0.913·13-s − 3.58i·17-s − 4i·19-s + (4.58 − 1.99i)25-s + 7.84i·29-s − 5.29·31-s + (0.417 + 1.99i)35-s − 7.84·37-s − 6·41-s + 7.16·43-s − 6.92i·47-s + 6.16·49-s + ⋯ |
| L(s) = 1 | + (−0.978 + 0.204i)5-s − 0.345i·7-s + 1.08i·11-s + 0.253·13-s − 0.868i·17-s − 0.917i·19-s + (0.916 − 0.399i)25-s + 1.45i·29-s − 0.950·31-s + (0.0705 + 0.338i)35-s − 1.28·37-s − 0.937·41-s + 1.09·43-s − 1.01i·47-s + 0.880·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092791199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092791199\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.18 - 0.456i)T \) |
| good | 7 | \( 1 + 0.913iT - 7T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 - 0.913T + 13T^{2} \) |
| 17 | \( 1 + 3.58iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 7.84iT - 29T^{2} \) |
| 31 | \( 1 + 5.29T + 31T^{2} \) |
| 37 | \( 1 + 7.84T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 7.58iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 7.16iT - 97T^{2} \) |
| 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
−8.687076525145626034533847247237, −7.73798484233679115927359868705, −7.01454309207103059599966613688, −6.83142312639007721758214686238, −5.31374102775454137755547242767, −4.74839102576623973182563187368, −3.83911447074284726719858156589, −3.08965076835682512648064748049, −1.89784122970184429528566597000, −0.43584334449993357908251174287,
0.957336320446411597133264084955, 2.29894247864117417237016901876, 3.62278397419270108024858030187, 3.84486003475757228115059896243, 5.09569694072526889435018610233, 5.84913683795682544708805031810, 6.56249041352521478958023101113, 7.62387153272583307716291957397, 8.192410910778425023805689647899, 8.701595230978895987997291269830
