L(s) = 1 | + (−1.23 − 2.13i)2-s + (−2.04 + 3.53i)4-s + (1.88 − 1.85i)7-s + 5.14·8-s + (−2.50 − 1.44i)11-s + 5.72·13-s + (−6.28 − 1.74i)14-s + (−2.25 − 3.91i)16-s + (−2.54 − 1.46i)17-s + (3.13 − 1.81i)19-s + 7.12i·22-s + (3.96 + 6.87i)23-s + (−7.06 − 12.2i)26-s + (2.69 + 10.4i)28-s + 5.38i·29-s + ⋯ |
L(s) = 1 | + (−0.872 − 1.51i)2-s + (−1.02 + 1.76i)4-s + (0.713 − 0.700i)7-s + 1.81·8-s + (−0.754 − 0.435i)11-s + 1.58·13-s + (−1.68 − 0.467i)14-s + (−0.564 − 0.978i)16-s + (−0.616 − 0.356i)17-s + (0.719 − 0.415i)19-s + 1.51i·22-s + (0.827 + 1.43i)23-s + (−1.38 − 2.40i)26-s + (0.509 + 1.97i)28-s + 1.00i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.111318477\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.111318477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.88 + 1.85i)T \) |
good | 2 | \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (2.50 + 1.44i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 17 | \( 1 + (2.54 + 1.46i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.13 + 1.81i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.96 - 6.87i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5.38iT - 29T^{2} \) |
| 31 | \( 1 + (-0.435 - 0.251i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.82 + 1.63i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 10.1iT - 43T^{2} \) |
| 47 | \( 1 + (-6.55 + 3.78i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.03 + 8.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.28 + 5.69i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (9.67 - 5.58i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.39 - 1.38i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.14iT - 71T^{2} \) |
| 73 | \( 1 + (0.284 - 0.492i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.58 + 7.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.0iT - 83T^{2} \) |
| 89 | \( 1 + (-1.76 - 3.05i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−9.173221484404584581926975873952, −8.634852470718415079609586910379, −7.85674989739556425589516064428, −7.11724327670893407139560246564, −5.71368972812526269865095216990, −4.65008115407491783201945257849, −3.63142396776227346156886948397, −2.94582902479892155462330868431, −1.62778356690398277997641283148, −0.798851438946441602934037778639,
1.02670731056210044515236417221, 2.47616826655530441287699107168, 4.18740755634870120691821038234, 5.13759758067142549980753318937, 5.88754797852690687062372280864, 6.48046219465254326760371297688, 7.51716556894770892432361904365, 8.092765722259806649849996460062, 8.776258231552475325188733107524, 9.213765660181414716160987122917