| L(s) = 1 | + i·2-s − 3i·3-s − 4-s + 3·6-s − i·8-s − 6·9-s − 2·11-s + 3i·12-s + 16-s − 4i·17-s − 6i·18-s + 6·19-s − 2i·22-s − 3i·23-s − 3·24-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.73i·3-s − 0.5·4-s + 1.22·6-s − 0.353i·8-s − 2·9-s − 0.603·11-s + 0.866i·12-s + 0.250·16-s − 0.970i·17-s − 1.41i·18-s + 1.37·19-s − 0.426i·22-s − 0.625i·23-s − 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3368947870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3368947870\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 + 9iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 7iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.136525531681965881330905095430, −7.47065033358485359870642212617, −7.24750453353937110079891465061, −6.29640746464056527390283525444, −5.61132289907456208918930119980, −4.90013013610484023793098619359, −3.40295091403241655453408914145, −2.45905658221863249662093823224, −1.30922243282634210104017822535, −0.11442797233236335246687668959,
1.81999077786162835530591016042, 3.18762517810985021672298877012, 3.58903939073236119838101542511, 4.47090707626934530569205143504, 5.37529989141844816863279257437, 5.69002718408736039276116007517, 7.23388523538313548028243484496, 8.182304880898049368889721690946, 8.935990719661259047961201329530, 9.551376475644920949317094753199
