| L(s) = 1 | + i·2-s − 4-s + (−2 − i)5-s − 2i·7-s − i·8-s + 3·9-s + (1 − 2i)10-s − 2i·13-s + 2·14-s + 16-s − 6i·17-s + 3i·18-s + 6·19-s + (2 + i)20-s − 4i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s − 0.755i·7-s − 0.353i·8-s + 9-s + (0.316 − 0.632i)10-s − 0.554i·13-s + 0.534·14-s + 0.250·16-s − 1.45i·17-s + 0.707i·18-s + 1.37·19-s + (0.447 + 0.223i)20-s − 0.834i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05890 - 0.249973i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05890 - 0.249973i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (2 + i)T \) |
| 37 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
| 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
−11.40506968222612717889534197365, −10.25113719858917368291542275203, −9.438090247624500579410071547213, −8.322965245049030687385456798168, −7.35568832889714045281533432589, −7.01158820365425114598423808349, −5.30465217903894502846767683368, −4.49338082846149928806726260801, −3.40786179572380380383719684032, −0.820739007894080673431905223777,
1.71098677704674363115611356244, 3.30426030742188220577888812198, 4.17325041782622047298026959064, 5.46016000559177642959206737587, 6.84419121247274177434054961507, 7.79721975732335667065088838795, 8.798930465226186592441211260912, 9.783773818615702590309723032008, 10.62908174788076414305464825761, 11.59081892644473436960668470110
