| L(s) = 1 | − 2.60i·3-s + 2.76·7-s − 3.76·9-s − 2.16i·11-s + (3.60 + 0.167i)13-s − 5.03i·19-s − 7.20i·21-s + 4.93i·23-s + 2.00i·27-s + 4.43·29-s − 3.37i·31-s − 5.63·33-s − 3.97·37-s + (0.434 − 9.37i)39-s − 1.66i·41-s + ⋯ |
| L(s) = 1 | − 1.50i·3-s + 1.04·7-s − 1.25·9-s − 0.653i·11-s + (0.998 + 0.0463i)13-s − 1.15i·19-s − 1.57i·21-s + 1.02i·23-s + 0.384i·27-s + 0.823·29-s − 0.605i·31-s − 0.981·33-s − 0.653·37-s + (0.0695 − 1.50i)39-s − 0.260i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.869529593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.869529593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.60 - 0.167i)T \) |
| good | 3 | \( 1 + 2.60iT - 3T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 2.16iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.03iT - 19T^{2} \) |
| 23 | \( 1 - 4.93iT - 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 + 3.37iT - 31T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + 1.66iT - 41T^{2} \) |
| 43 | \( 1 + 9.80iT - 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 8.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 7.10T + 67T^{2} \) |
| 71 | \( 1 - 16.1iT - 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 + 7.94iT - 89T^{2} \) |
| 97 | \( 1 + 0.462T + 97T^{2} \) |
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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
−8.917807323210430565201572552718, −8.548629999185352441238133938774, −7.62439436118160580598730846039, −7.10140129864630006619717350447, −6.11510100879633329545021370379, −5.44011197847969232722046947739, −4.20475660234132530838171861254, −2.88503909316830081656311502494, −1.76929483318153897613428876280, −0.861118426789844080147037835151,
1.57265476781713916319366719910, 3.06251818159546009512284687442, 4.12623193425761556897203217489, 4.64215387545912438470877879344, 5.49218529949566943108804169564, 6.45794150738013136228855173368, 7.72771109094423583045640594547, 8.503031171865671666568546971927, 9.085844849928913206495848568264, 10.19611259067147085927296914357
