| L(s) = 1 | − i·2-s − 1.41i·3-s − 4-s − 1.41·6-s + i·8-s + 0.999·9-s − 2·11-s + 1.41i·12-s + 16-s + 1.41i·17-s − 0.999i·18-s + 7.07·19-s + 2i·22-s − 4i·23-s + 1.41·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.816i·3-s − 0.5·4-s − 0.577·6-s + 0.353i·8-s + 0.333·9-s − 0.603·11-s + 0.408i·12-s + 0.250·16-s + 0.342i·17-s − 0.235i·18-s + 1.62·19-s + 0.426i·22-s − 0.834i·23-s + 0.288·24-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
| \(L(1)\) |
\(\approx\) |
\(1.565702899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565702899\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41iT - 3T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 1.41iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 9.89iT - 97T^{2} \) |
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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
−8.544791413116244643192603271116, −7.86434443042299580411976907119, −7.20410680884697623109581105500, −6.34895607083798738812922741562, −5.39703501961417961774565221021, −4.58339779528212415410226986761, −3.54127434048832595028844361001, −2.59313551648394066405374158158, −1.66740413169707232129257630282, −0.59385065510202724474665624398,
1.24293490218134664069690712325, 2.93509116022770452964376302475, 3.70793225680344010697874165662, 4.81211703437517507733717847299, 5.13863499542476217145821917821, 6.10892993486972300448245676158, 7.09732980506389142922069560610, 7.64050759622656906626051827401, 8.492834349754692917693835355568, 9.295494188131052750373320817051
