| L(s) = 1 | + 1.73i·2-s − 0.999·4-s + 2i·7-s + 1.73i·8-s − 3.46·11-s − i·13-s − 3.46·14-s − 5·16-s − 6.92i·17-s − 2·19-s − 5.99i·22-s − 6.92i·23-s + 1.73·26-s − 1.99i·28-s − 6.92·29-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s + 0.755i·7-s + 0.612i·8-s − 1.04·11-s − 0.277i·13-s − 0.925·14-s − 1.25·16-s − 1.68i·17-s − 0.458·19-s − 1.27i·22-s − 1.44i·23-s + 0.339·26-s − 0.377i·28-s − 1.28·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7018329913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7018329913\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 17 | \( 1 + 6.92iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 10.3iT - 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.615513087020081902334949504462, −7.78896175465236078946199135918, −7.22121478306033271041096518970, −6.49061040860035439941711849030, −5.55759045154070788979653439559, −5.24658863438574082178813398988, −4.31311113416572596671201352315, −2.79185469229515332739538860600, −2.28625621297087126292755738529, −0.21417575453941567693903686781,
1.27491567267923539406844613668, 2.08159230916042271230957987179, 3.13777105191266907660845842579, 3.88969032024867705437802038694, 4.55106774943340329366770566473, 5.74710096825923939908363690228, 6.47485109495983676461380009415, 7.57820437639456591807855122533, 7.919435399262465262133380256516, 9.169846804498099158706982574420
