| L(s) = 1 | − 2.64·7-s − 0.913i·11-s + 9.39i·23-s − 5·25-s − 6.06i·29-s + 10.5·37-s + 5.29·43-s + 7.00·49-s − 14.5i·53-s − 4·67-s − 7.57i·71-s + 2.41i·77-s − 8·79-s − 17.8i·107-s + 10.5·109-s + ⋯ |
| L(s) = 1 | − 0.999·7-s − 0.275i·11-s + 1.95i·23-s − 25-s − 1.12i·29-s + 1.73·37-s + 0.806·43-s + 49-s − 1.99i·53-s − 0.488·67-s − 0.898i·71-s + 0.275i·77-s − 0.900·79-s − 1.72i·107-s + 1.01·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260435919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260435919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 0.913iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 9.39iT - 23T^{2} \) |
| 29 | \( 1 + 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 7.57iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−8.224244665114661870819067949668, −7.62217031969878489783457268502, −6.88254815595460782790593223727, −5.94627022954080182698152000351, −5.66264349615736164531706219180, −4.40984029277064941127044985049, −3.66518633587577326443341409593, −2.91479445927530436885369063196, −1.84702220210497660970332644134, −0.46238945665516349089101824057,
0.866804351346218679343491996311, 2.30648582968056149505582095831, 3.00018468481403763734614341769, 4.03868331006483467572529299005, 4.65171279414521176794323759818, 5.81046631117846668791549474875, 6.27736218972327754302860668405, 7.08522623573213825804644404311, 7.73970599116239742096300240895, 8.675795360975083186899413576782
