| L(s) = 1 | + i·2-s − 2.73·3-s + 4-s + 0.267i·5-s − 2.73i·6-s + 3.46·7-s + 3i·8-s + 4.46·9-s − 0.267·10-s − 4.73·11-s − 2.73·12-s + 3.46i·13-s + 3.46i·14-s − 0.732i·15-s − 16-s − 3.73i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.57·3-s + 0.5·4-s + 0.119i·5-s − 1.11i·6-s + 1.30·7-s + 1.06i·8-s + 1.48·9-s − 0.0847·10-s − 1.42·11-s − 0.788·12-s + 0.960i·13-s + 0.925i·14-s − 0.189i·15-s − 0.250·16-s − 0.905i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.012446746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.012446746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 0.267iT - 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.73iT - 17T^{2} \) |
| 19 | \( 1 - 1.26iT - 19T^{2} \) |
| 23 | \( 1 - 0.732iT - 23T^{2} \) |
| 29 | \( 1 + 0.267iT - 29T^{2} \) |
| 31 | \( 1 - 8.19iT - 31T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 13.4iT - 59T^{2} \) |
| 61 | \( 1 - 4.26iT - 61T^{2} \) |
| 67 | \( 1 + 7.26T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 3.80iT - 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 + 9.19iT - 89T^{2} \) |
| 97 | \( 1 - 4.26iT - 97T^{2} \) |
| 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
−10.40102916588257314398811562157, −8.923614178343520409072097006522, −7.987753852027107185227311103334, −7.24663218101769055747203687908, −6.69603987939780264618397632846, −5.66247033605941753601376455223, −5.13724194571181897833413008569, −4.58494218339222797341409458344, −2.65202006381937132784143449785, −1.39968664815450180048687224278,
0.52298687637646507111021569232, 1.68786882650733228046720062978, 2.85409343712370770562978503373, 4.32246500041763352875938103132, 5.13716901765607313895522057396, 5.75616275666468329116719178157, 6.64914319383755486334872601380, 7.69767107461360431902279112858, 8.206498459378218816360258455863, 9.751542762768407012333377264718
