| L(s) = 1 | + (−1 + 2i)5-s + 4i·7-s + 4i·13-s + 8·19-s − 4i·23-s + (−3 − 4i)25-s − 6·29-s − 8·31-s + (−8 − 4i)35-s + 4i·37-s − 6·41-s + 4i·43-s − 4i·47-s − 9·49-s + 12i·53-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s + 1.51i·7-s + 1.10i·13-s + 1.83·19-s − 0.834i·23-s + (−0.600 − 0.800i)25-s − 1.11·29-s − 1.43·31-s + (−1.35 − 0.676i)35-s + 0.657i·37-s − 0.937·41-s + 0.609i·43-s − 0.583i·47-s − 1.28·49-s + 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.101060931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.101060931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.610342966639099694925149433255, −9.226058171897001814659227291784, −8.265117223089157239411512886960, −7.41144368419041654986731398540, −6.64504622952495083972104400827, −5.79360653219560157313105291554, −4.96621721210875437351677184952, −3.71599304262311115499967452216, −2.85350458962345062408189968866, −1.87322452308456859788832528505,
0.45336944881666730517742830952, 1.49057005351059371121255204327, 3.41851728508801929838304482548, 3.85664316549330218630053728810, 5.10987422315487353294961914387, 5.57248287375360996119390763337, 7.13402074126162639589090427692, 7.52342135744687534022378169885, 8.209151443680084889922028916982, 9.333477879231934759220167449011
