L(s) = 1 | − 3.69i·5-s − 7-s + 3.21i·11-s + 5.08i·13-s − 0.616·17-s + 4.48i·19-s + 1.38·23-s − 8.67·25-s + 5.67i·29-s − 6.91·31-s + 3.69i·35-s − 6.91i·37-s + 0.616·41-s + 7.99i·43-s + 4.97·47-s + ⋯ |
L(s) = 1 | − 1.65i·5-s − 0.377·7-s + 0.968i·11-s + 1.40i·13-s − 0.149·17-s + 1.02i·19-s + 0.288·23-s − 1.73·25-s + 1.05i·29-s − 1.24·31-s + 0.625i·35-s − 1.13i·37-s + 0.0963·41-s + 1.21i·43-s + 0.725·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.126042947\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.126042947\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
good | 5 | \( 1 + 3.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.21iT - 11T^{2} \) |
| 13 | \( 1 - 5.08iT - 13T^{2} \) |
| 17 | \( 1 + 0.616T + 17T^{2} \) |
| 19 | \( 1 - 4.48iT - 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 - 5.67iT - 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 + 6.91iT - 37T^{2} \) |
| 41 | \( 1 - 0.616T + 41T^{2} \) |
| 43 | \( 1 - 7.99iT - 43T^{2} \) |
| 47 | \( 1 - 4.97T + 47T^{2} \) |
| 53 | \( 1 + 4.48iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 9.56iT - 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−9.072797301994492848715956488567, −8.864460418877149732174205961555, −7.71774720055810666564778794631, −7.04815395853457220323928674080, −6.00892829883466380860513786802, −5.17502964818924723366303007415, −4.42095971256475915147102920408, −3.77156675128064471165304416738, −2.11365124213734510672181314321, −1.28173118716014851472324709207,
0.41692222181428674877801658412, 2.36695148698659267718687781819, 3.12283678709362070655620515940, 3.68447421205840334847315931074, 5.14282208367968470220512577359, 6.01979869289415143248811714570, 6.60450417997915801562699012772, 7.41687699403819076861153776943, 8.073545790481856172776610107305, 9.073322301636243947410794176429