| L(s) = 1 | − 1.91·3-s − 0.910i·5-s + 3.26·7-s + 0.648·9-s − 2.79·11-s + 1.79i·13-s + 1.73i·15-s − 4.96i·17-s + 2.96i·19-s − 6.22·21-s − 1.35i·23-s + 4.17·25-s + 4.49·27-s − 5.69i·29-s + 6.05i·31-s + ⋯ |
| L(s) = 1 | − 1.10·3-s − 0.407i·5-s + 1.23·7-s + 0.216·9-s − 0.843·11-s + 0.498i·13-s + 0.448i·15-s − 1.20i·17-s + 0.681i·19-s − 1.35·21-s − 0.281i·23-s + 0.834·25-s + 0.864·27-s − 1.05i·29-s + 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0186 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9530819247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9530819247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (0.113 + 6.08i)T \) |
| good | 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 + 0.910iT - 5T^{2} \) |
| 7 | \( 1 - 3.26T + 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 1.79iT - 13T^{2} \) |
| 17 | \( 1 + 4.96iT - 17T^{2} \) |
| 19 | \( 1 - 2.96iT - 19T^{2} \) |
| 23 | \( 1 + 1.35iT - 23T^{2} \) |
| 29 | \( 1 + 5.69iT - 29T^{2} \) |
| 31 | \( 1 - 6.05iT - 31T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 - 0.851iT - 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 5.70T + 53T^{2} \) |
| 59 | \( 1 + 0.342iT - 59T^{2} \) |
| 61 | \( 1 + 2.28iT - 61T^{2} \) |
| 67 | \( 1 - 3.35T + 67T^{2} \) |
| 71 | \( 1 + 9.19T + 71T^{2} \) |
| 73 | \( 1 - 15.9T + 73T^{2} \) |
| 79 | \( 1 - 9.06iT - 79T^{2} \) |
| 83 | \( 1 - 6.73T + 83T^{2} \) |
| 89 | \( 1 + 4.26iT - 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
| 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
−8.658918684372926778233663266611, −8.047627370310051256824722864194, −7.22675010011099067329520923387, −6.36496100421093328730148944137, −5.37643988085632758710025278755, −5.02727137532980408814006211856, −4.32005807529156243570263453435, −2.84917227492470978199207645583, −1.65955596590029193385496100707, −0.43883955364637659747313951281,
1.08245722826055031530549891189, 2.33491743564399133179706082312, 3.45941552265599163524916632800, 4.82212410713180198640900369695, 5.08531224368804500175582870480, 6.01269694304090265456182027451, 6.69803327274867774266819538850, 7.71447201157868388918947343349, 8.216335731538895051842139586716, 9.090312040375986516617250947378
