L(s) = 1 | + (0.707 − 0.707i)3-s + (−2.82 − 2.82i)7-s + 1.99i·9-s − 5.19i·11-s + (−2.44 − 2.44i)13-s + (−3.67 + 3.67i)17-s + 1.73·19-s − 4.00·21-s + (−4.24 + 4.24i)23-s + (3.53 + 3.53i)27-s + 6i·29-s + 3.46i·31-s + (−3.67 − 3.67i)33-s − 3.46·39-s − 3·41-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−1.06 − 1.06i)7-s + 0.666i·9-s − 1.56i·11-s + (−0.679 − 0.679i)13-s + (−0.891 + 0.891i)17-s + 0.397·19-s − 0.872·21-s + (−0.884 + 0.884i)23-s + (0.680 + 0.680i)27-s + 1.11i·29-s + 0.622i·31-s + (−0.639 − 0.639i)33-s − 0.554·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2384585615\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2384585615\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.82 + 2.82i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + (2.44 + 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.67 - 3.67i)T - 17iT^{2} \) |
| 19 | \( 1 - 1.73T + 19T^{2} \) |
| 23 | \( 1 + (4.24 - 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + (2.82 - 2.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.24 - 4.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.34 + 7.34i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + (4.94 + 4.94i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (11.0 + 11.0i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-2.12 + 2.12i)T - 83iT^{2} \) |
| 89 | \( 1 - 3iT - 89T^{2} \) |
| 97 | \( 1 + (4.89 - 4.89i)T - 97iT^{2} \) |
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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
−8.858740667711278582659464329355, −8.062233410966065524528287023564, −7.45485212098929404716440684579, −6.59091829940006503899221705769, −5.83236421905734433981334098775, −4.75854602105141345118225159476, −3.50038738971508459496419983257, −3.04553495819540601820870495493, −1.56731301217924907622284017238, −0.081474071634927170735122506685,
2.20620660610336534783670515054, 2.77480431698165888056412435945, 4.07256841897460576515650563279, 4.69525858679575286985239336140, 5.90730997550882402611996027842, 6.67981069131978467015572039665, 7.32512489157396519780023361547, 8.542705544660088959485285658704, 9.304227682241307465134178192161, 9.658512572779348690002907884388