| L(s) = 1 | − 1.38i·3-s + 1.38i·5-s − 2.44·7-s + 1.07·9-s + 1.05i·11-s + i·13-s + 1.92·15-s + 6.96·17-s − 0.408i·19-s + 3.38i·21-s + 3.57·23-s + 3.07·25-s − 5.65i·27-s − 4.34i·29-s + 5.74·31-s + ⋯ |
| L(s) = 1 | − 0.800i·3-s + 0.620i·5-s − 0.923·7-s + 0.359·9-s + 0.318i·11-s + 0.277i·13-s + 0.496·15-s + 1.68·17-s − 0.0936i·19-s + 0.738i·21-s + 0.745·23-s + 0.615·25-s − 1.08i·27-s − 0.807i·29-s + 1.03·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53249 - 0.201757i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53249 - 0.201757i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 1.38iT - 3T^{2} \) |
| 5 | \( 1 - 1.38iT - 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.05iT - 11T^{2} \) |
| 17 | \( 1 - 6.96T + 17T^{2} \) |
| 19 | \( 1 + 0.408iT - 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 + 4.34iT - 29T^{2} \) |
| 31 | \( 1 - 5.74T + 31T^{2} \) |
| 37 | \( 1 - 7.49iT - 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 - 2.27iT - 43T^{2} \) |
| 47 | \( 1 - 4.33T + 47T^{2} \) |
| 53 | \( 1 + 0.647iT - 53T^{2} \) |
| 59 | \( 1 + 3.82iT - 59T^{2} \) |
| 61 | \( 1 - 4.80iT - 61T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 + 7.66T + 71T^{2} \) |
| 73 | \( 1 - 7.68T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.88T + 89T^{2} \) |
| 97 | \( 1 + 3.30T + 97T^{2} \) |
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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
−9.991649215448975685477592312547, −9.623096955855276854361724526744, −8.276337179020028046219129401806, −7.46225670263793541587146945535, −6.72922004679251919875231502353, −6.16216125041544756888513114640, −4.84845369379135013040715596789, −3.51577932665919583871093361202, −2.60519516249526934051850021928, −1.11164674692079799674910491433,
1.04102091931679645785355241802, 3.02316352988011125317001744781, 3.79776813350475827587588400830, 4.93014362062895838117856658312, 5.64509365811596918764833513692, 6.78556664107993911600837102878, 7.74187039951352558485880210878, 8.824615105005562120009288337006, 9.441441301207292440237919241209, 10.21108408556250339046305902250
