| L(s) = 1 | + 2·7-s + 5·11-s + 5·17-s − 5·19-s + 6·23-s + 4·29-s + 10·31-s − 10·37-s − 5·41-s + 4·43-s − 8·47-s − 3·49-s + 10·53-s + 10·61-s + 3·67-s + 5·73-s + 10·77-s + 10·79-s + 83-s + 9·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 10·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.50·11-s + 1.21·17-s − 1.14·19-s + 1.25·23-s + 0.742·29-s + 1.79·31-s − 1.64·37-s − 0.780·41-s + 0.609·43-s − 1.16·47-s − 3/7·49-s + 1.37·53-s + 1.28·61-s + 0.366·67-s + 0.585·73-s + 1.13·77-s + 1.12·79-s + 0.109·83-s + 0.953·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.916·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.124864420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.124864420\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.19522179110595, −15.38899406508745, −14.95127524991255, −14.42976399052173, −14.03306079761115, −13.41747932393694, −12.60982261333097, −12.10453324166813, −11.64853389395178, −11.12510984778404, −10.31409514157481, −9.969413713718265, −9.089580979739027, −8.565204257419526, −8.168414593868437, −7.302859154774937, −6.604420509645969, −6.292436841546783, −5.181427421849367, −4.840114347147274, −3.950908341051119, −3.391998792280796, −2.416175606331794, −1.482807897577456, −0.8665792315333439,
0.8665792315333439, 1.482807897577456, 2.416175606331794, 3.391998792280796, 3.950908341051119, 4.840114347147274, 5.181427421849367, 6.292436841546783, 6.604420509645969, 7.302859154774937, 8.168414593868437, 8.565204257419526, 9.089580979739027, 9.969413713718265, 10.31409514157481, 11.12510984778404, 11.64853389395178, 12.10453324166813, 12.60982261333097, 13.41747932393694, 14.03306079761115, 14.42976399052173, 14.95127524991255, 15.38899406508745, 16.19522179110595
