| L(s) = 1 | + i·2-s − 1.23i·3-s − 4-s + 1.23·6-s + i·7-s − i·8-s + 1.47·9-s + 11-s + 1.23i·12-s + 3.23i·13-s − 14-s + 16-s + 2.47i·17-s + 1.47i·18-s + 7.23·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.713i·3-s − 0.5·4-s + 0.504·6-s + 0.377i·7-s − 0.353i·8-s + 0.490·9-s + 0.301·11-s + 0.356i·12-s + 0.897i·13-s − 0.267·14-s + 0.250·16-s + 0.599i·17-s + 0.346i·18-s + 1.66·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.921116968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.921116968\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23iT - 3T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94iT - 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−8.315317060789107727071122532170, −7.895148529513038443314393939887, −6.96590006657899645886676076439, −6.62149593377679849768984600009, −5.83791708322714542717812600052, −4.93848841914833130457038765595, −4.19052650269960394029648910069, −3.18839589877662855690976497654, −1.94986555002684702607324266923, −1.05011150078291730518513109930,
0.68787314258593422075043271021, 1.74718992675156946781785819827, 3.08925172179953132045572932829, 3.58791658955398145514465800048, 4.41832124569640023645857894056, 5.23374207378731136532822275554, 5.78571916836636751498554114074, 7.26886995833488501868615058436, 7.44054862136265175796281414264, 8.573752991397991703432440182547
