| L(s) = 1 | + i·2-s − 4-s + (2 + i)5-s − 2i·7-s − i·8-s + (−1 + 2i)10-s + 4·11-s − 2i·13-s + 2·14-s + 16-s + 6i·17-s − 4·19-s + (−2 − i)20-s + 4i·22-s + (3 + 4i)25-s + 2·26-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 + (−0.316 + 0.632i)10-s + 1.20·11-s − 0.554i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s − 0.917·19-s + (−0.447 − 0.223i)20-s + 0.852i·22-s + (0.600 + 0.800i)25-s + 0.392·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.209552966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.209552966\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−8.761517094717355604273970603784, −8.260734868488792664472413144114, −7.20362365466727305038331240103, −6.59253756002942286688827595138, −6.10457356830804435143921625847, −5.25254825207630439003445249599, −4.14589273795814278155203325111, −3.57940189006391263707356431859, −2.17711126895890250240907946992, −1.04763402532140809353572430221,
0.898290197758227478234619723578, 2.00862243358654684701717613574, 2.65625198489205159502731350559, 3.89449677208577056699080305377, 4.72682337709213843771622550473, 5.45951289154088555721480817904, 6.33492401798080683605081260243, 6.97318972604869314635526539652, 8.299821899688487444811615631045, 8.931267458127281932893317209283
