| L(s) = 1 | + 3-s + 2i·5-s + 2i·7-s + 9-s + 4i·11-s + (−3 − 2i)13-s + 2i·15-s + 6·17-s − 2i·19-s + 2i·21-s − 4·23-s + 25-s + 27-s + 6·29-s − 2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894i·5-s + 0.755i·7-s + 0.333·9-s + 1.20i·11-s + (−0.832 − 0.554i)13-s + 0.516i·15-s + 1.45·17-s − 0.458i·19-s + 0.436i·21-s − 0.834·23-s + 0.200·25-s + 0.192·27-s + 1.11·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35354 + 0.724393i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35354 + 0.724393i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
| 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
−12.23416523760341517012630761573, −10.65586757290161388459012614812, −9.967718907482976454477574665922, −9.127673101909468779132099900692, −7.82051491543289708817066496019, −7.22929784585604966633918591841, −5.95750109635527664408213966840, −4.70885138345701211341794423163, −3.18417782888873328668405113284, −2.23185768881719592161745872275,
1.19324530195009471651560356750, 3.11774083631319730255891746586, 4.29735082485778753658538410279, 5.42143325931896951905538082359, 6.76470848570786728670230976514, 7.983005416588350245760703232595, 8.518645322848007945416252594660, 9.710000530282027186185716178835, 10.35102684008798778077439152506, 11.73424193741003206223035966760
