| L(s) = 1 | − 1.75·5-s + 6.54·11-s + 3.60i·13-s − 1.92·25-s + 10.1i·41-s − 12.4i·43-s + 9.33i·47-s + 7·49-s − 11.4·55-s − 0.469·59-s − 7.21i·61-s − 6.31i·65-s + 4.92i·71-s − 10.3·79-s + 12.6·83-s + ⋯ |
| L(s) = 1 | − 0.783·5-s + 1.97·11-s + 0.999i·13-s − 0.385·25-s + 1.58i·41-s − 1.90i·43-s + 1.36i·47-s + 49-s − 1.54·55-s − 0.0611·59-s − 0.923i·61-s − 0.783i·65-s + 0.584i·71-s − 1.16·79-s + 1.38·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.610963616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.610963616\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - 3.60iT \) |
| good | 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6.54T + 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 10.1iT - 41T^{2} \) |
| 43 | \( 1 + 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 9.33iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 0.469T + 59T^{2} \) |
| 61 | \( 1 + 7.21iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 4.92iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 18.3iT - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
| 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
−8.688411147040547645619033218197, −7.911514083741683628959981743247, −7.03654938954395075611960030923, −6.58328521467525885798750089375, −5.76652244305961268960762223778, −4.54745779195063722675496167046, −4.05882672577957391329777186202, −3.38451090748335271038203374638, −2.04279989404844821042899662250, −1.03859448182280511821680306617,
0.57908950013993733940537032279, 1.69373129254155630524216484539, 3.02718277677129046206737563460, 3.83713538813522687527506730626, 4.33837134236243811836649847689, 5.46071272857525897833891106091, 6.20582377471527069653618402735, 7.00106327192841045722532708428, 7.62263519722959950055782810397, 8.438492829671094526413836331671
