L(s) = 1 | − 2.24i·3-s − 3.30i·5-s + 7-s − 2.05·9-s − 0.941i·11-s + 5.19i·13-s − 7.43·15-s − 6.49·17-s + 1.75i·19-s − 2.24i·21-s − 5.55·23-s − 5.94·25-s − 2.11i·27-s − 2.49i·29-s − 6.61·31-s + ⋯ |
L(s) = 1 | − 1.29i·3-s − 1.47i·5-s + 0.377·7-s − 0.686·9-s − 0.283i·11-s + 1.43i·13-s − 1.92·15-s − 1.57·17-s + 0.401i·19-s − 0.490i·21-s − 1.15·23-s − 1.18·25-s − 0.407i·27-s − 0.463i·29-s − 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8210541677\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8210541677\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 2.24iT - 3T^{2} \) |
| 5 | \( 1 + 3.30iT - 5T^{2} \) |
| 11 | \( 1 + 0.941iT - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 1.75iT - 19T^{2} \) |
| 23 | \( 1 + 5.55T + 23T^{2} \) |
| 29 | \( 1 + 2.49iT - 29T^{2} \) |
| 31 | \( 1 + 6.61T + 31T^{2} \) |
| 37 | \( 1 - 4.61iT - 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 0.443iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.117T + 73T^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−8.690385851138362765285194158043, −8.112438193216527576040576404560, −7.18131095739526657975885297645, −6.50659130930162558319155186282, −5.63096721205063827585379512560, −4.63980008495423788641195537033, −3.95652187145484919608445073356, −2.01579660570633269589244517561, −1.69394394341835347945538518818, −0.28280344569666110637029591097,
2.17781932329443526265331306934, 3.13540035918723197363166637823, 3.88744148353217125312799752818, 4.78456056387388932699567483604, 5.64849264847664806764774974940, 6.60253426121619506032576345404, 7.37055406628838134335610586159, 8.242748694889640576786288336079, 9.187928846832657779871190105810, 9.923611232975024842121433550637