| L(s) = 1 | + i·5-s + 0.585i·7-s − 5.41·11-s − 0.585·13-s − 6.82i·17-s + 0.828·23-s − 25-s − 6i·29-s − 10.4i·31-s − 0.585·35-s + 5.07·37-s + 3.07i·41-s − 1.17i·43-s − 5.65·47-s + 6.65·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 0.221i·7-s − 1.63·11-s − 0.162·13-s − 1.65i·17-s + 0.172·23-s − 0.200·25-s − 1.11i·29-s − 1.88i·31-s − 0.0990·35-s + 0.833·37-s + 0.479i·41-s − 0.178i·43-s − 0.825·47-s + 0.950·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8521020765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8521020765\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| good | 7 | \( 1 - 0.585iT - 7T^{2} \) |
| 11 | \( 1 + 5.41T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 + 6.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 - 5.07T + 37T^{2} \) |
| 41 | \( 1 - 3.07iT - 41T^{2} \) |
| 43 | \( 1 + 1.17iT - 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 6.82iT - 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 - 7.17T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 + 2.48iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−9.662337011780029752071910847615, −8.326735264189336075440557593943, −7.70592866049216558615303575379, −7.00949458106151455618334243107, −5.90921358044068909375144391480, −5.20555527959876291940503360636, −4.26330982678663996564535477361, −2.87781869918281055013556588308, −2.36924244560887649771519773585, −0.33777597376407490990255275440,
1.40958643605283253068621518841, 2.68623303248484705423205622993, 3.75469928657225169008237401970, 4.85873886610452850591929924530, 5.47246576712210034803522664290, 6.48144415644044816824336908772, 7.46532338857206512988206168615, 8.197477064952409971209784605130, 8.793542294235827558060716248978, 9.840675879227946458140693846234
