| L(s) = 1 | − 3-s − 3.75·5-s + 4.75·7-s + 9-s − 4.36·11-s + 13-s + 3.75·15-s + 6.12·17-s − 4.75·21-s − 3.75·23-s + 9.12·25-s − 27-s + 5.38·29-s + 7.36·31-s + 4.36·33-s − 17.8·35-s − 7.12·37-s − 39-s − 3.51·41-s − 0.758·43-s − 3.75·45-s − 6·47-s + 15.6·49-s − 6.12·51-s + 5.14·53-s + 16.4·55-s + 9.27·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.68·5-s + 1.79·7-s + 0.333·9-s − 1.31·11-s + 0.277·13-s + 0.970·15-s + 1.48·17-s − 1.03·21-s − 0.783·23-s + 1.82·25-s − 0.192·27-s + 1.00·29-s + 1.32·31-s + 0.760·33-s − 3.02·35-s − 1.17·37-s − 0.160·39-s − 0.549·41-s − 0.115·43-s − 0.560·45-s − 0.875·47-s + 2.23·49-s − 0.858·51-s + 0.707·53-s + 2.21·55-s + 1.20·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.271602584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271602584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 - 4.75T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 23 | \( 1 + 3.75T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 3.51T + 41T^{2} \) |
| 43 | \( 1 + 0.758T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 5.14T + 53T^{2} \) |
| 59 | \( 1 - 9.27T + 59T^{2} \) |
| 61 | \( 1 + 8.51T + 61T^{2} \) |
| 67 | \( 1 - 1.36T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 4.51T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 4.90T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
| 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
−7.87761097544322484149221718923, −7.37978434754927885053756612229, −6.46162484748277189807391197975, −5.33690174097743331251494902255, −5.05824103254662511756596457328, −4.36648975279828454686234965199, −3.64575463333201338019714295700, −2.72362863093109388166015023493, −1.51100052800315348506608389244, −0.60310524256666930289876194525,
0.60310524256666930289876194525, 1.51100052800315348506608389244, 2.72362863093109388166015023493, 3.64575463333201338019714295700, 4.36648975279828454686234965199, 5.05824103254662511756596457328, 5.33690174097743331251494902255, 6.46162484748277189807391197975, 7.37978434754927885053756612229, 7.87761097544322484149221718923
