| L(s) = 1 | + 2.82i·5-s − i·7-s + 1.41·11-s + 2·13-s − 7.07·23-s − 3.00·25-s + 9.89i·29-s + 4i·31-s + 2.82·35-s + 2.82i·41-s − 4i·43-s − 2.82·47-s − 49-s + 4.24i·53-s + 4.00i·55-s + ⋯ |
| L(s) = 1 | + 1.26i·5-s − 0.377i·7-s + 0.426·11-s + 0.554·13-s − 1.47·23-s − 0.600·25-s + 1.83i·29-s + 0.718i·31-s + 0.478·35-s + 0.441i·41-s − 0.609i·43-s − 0.412·47-s − 0.142·49-s + 0.582i·53-s + 0.539i·55-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
| \(L(1)\) |
\(\approx\) |
\(1.407258760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.407258760\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 9.89iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 2.82iT - 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−8.598826603626341235357504142876, −7.938285512723348471778007896290, −7.01020179448498244962021493745, −6.68625101425017635509286052502, −5.90379231733636370091699555808, −4.97208474739099251564042827583, −3.85483962267815838752224897382, −3.40291797692853083245820308844, −2.39700734866902986619606575740, −1.31391433769813570365900192226,
0.41360937206465012608049335155, 1.54272228800375929072934093144, 2.45778872708684952647701927580, 3.82659889905347132719623806491, 4.29981104990853312764551742033, 5.24309889351418067578033301150, 5.92364246754254473711398738835, 6.53209000061440518093682927216, 7.74055155791217289352658804994, 8.237570906598917130704719346900
