L(s) = 1 | − i·3-s − 4i·7-s − 9-s + 2i·13-s + 6i·17-s − 4·21-s + 4i·23-s + i·27-s − 2·29-s + 8·31-s + 6i·37-s + 2·39-s − 6·41-s + 12i·43-s − 12i·47-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s − 0.872·21-s + 0.834i·23-s + 0.192i·27-s − 0.371·29-s + 1.43·31-s + 0.986i·37-s + 0.320·39-s − 0.937·41-s + 1.82i·43-s − 1.75i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.458389132\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458389132\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 18iT - 97T^{2} \) |
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
−8.142077053542090452937623860307, −7.65676285020669509348833128935, −6.79862364274803939072699091981, −6.45327649735688479073262657189, −5.48630693398022378612948356195, −4.44752329369589077891847631620, −3.89788981064797320116317041147, −2.99887982735259436365839832554, −1.71845042998606194333522318367, −1.04215264565849438628334219935,
0.44389073045437874445937500937, 2.13772497377862640766844385390, 2.77990118909957944309262235076, 3.57002898429722302625340389941, 4.80260793503639545760101810922, 5.14066850736440969895472032040, 5.97055632529444397149678969396, 6.62577972291327813069864671777, 7.66528511008111724422641795556, 8.328983506297592998024417676885