| L(s) = 1 | − 2·7-s − 4·11-s + 13-s − 17-s − 2·19-s − 7·23-s + 6·29-s − 4·31-s + 11·37-s + 9·41-s − 6·43-s + 4·47-s − 3·49-s + 53-s − 59-s + 5·61-s + 2·67-s + 3·71-s + 4·73-s + 8·77-s + 6·79-s + 11·83-s − 2·91-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 0.277·13-s − 0.242·17-s − 0.458·19-s − 1.45·23-s + 1.11·29-s − 0.718·31-s + 1.80·37-s + 1.40·41-s − 0.914·43-s + 0.583·47-s − 3/7·49-s + 0.137·53-s − 0.130·59-s + 0.640·61-s + 0.244·67-s + 0.356·71-s + 0.468·73-s + 0.911·77-s + 0.675·79-s + 1.20·83-s − 0.209·91-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570144424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570144424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−12.37305294243143, −12.22235285891122, −11.37484015657940, −11.14659103988763, −10.59087757608719, −10.13709784657013, −9.840363656382313, −9.327285184407054, −8.849865886198205, −8.164775135262654, −8.012292272453254, −7.491068610336821, −6.882077352486810, −6.356313091025778, −6.008265791275182, −5.611249272022979, −4.885872827090483, −4.510032118058629, −3.885831585428073, −3.450964613947608, −2.762861670226499, −2.365221897552743, −1.890180864505582, −0.8884501610280653, −0.3814492336321048,
0.3814492336321048, 0.8884501610280653, 1.890180864505582, 2.365221897552743, 2.762861670226499, 3.450964613947608, 3.885831585428073, 4.510032118058629, 4.885872827090483, 5.611249272022979, 6.008265791275182, 6.356313091025778, 6.882077352486810, 7.491068610336821, 8.012292272453254, 8.164775135262654, 8.849865886198205, 9.327285184407054, 9.840363656382313, 10.13709784657013, 10.59087757608719, 11.14659103988763, 11.37484015657940, 12.22235285891122, 12.37305294243143
