| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·5-s − 3·6-s − 7-s + 8-s + 6·9-s − 3·10-s + 11-s − 3·12-s − 14-s + 9·15-s + 16-s − 8·17-s + 6·18-s − 3·20-s + 3·21-s + 22-s − 3·23-s − 3·24-s + 4·25-s − 9·27-s − 28-s + 9·30-s − 2·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.34·5-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 0.948·10-s + 0.301·11-s − 0.866·12-s − 0.267·14-s + 2.32·15-s + 1/4·16-s − 1.94·17-s + 1.41·18-s − 0.670·20-s + 0.654·21-s + 0.213·22-s − 0.625·23-s − 0.612·24-s + 4/5·25-s − 1.73·27-s − 0.188·28-s + 1.64·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.73471084285795, −15.18165707050189, −14.77609115295171, −13.75372841864295, −13.36286584898565, −12.59633584953222, −12.32910848167505, −11.87585623497952, −11.22243794268863, −11.03129783779673, −10.61162418435983, −9.701840751104292, −9.116877787900486, −8.204429526010892, −7.614104473942300, −6.925593117426449, −6.575170592853163, −6.048246228009258, −5.359706767669320, −4.677508239844083, −4.139198767404653, −3.899505009046882, −2.800886026185312, −1.844820049841074, −0.7205181920548795, 0,
0.7205181920548795, 1.844820049841074, 2.800886026185312, 3.899505009046882, 4.139198767404653, 4.677508239844083, 5.359706767669320, 6.048246228009258, 6.575170592853163, 6.925593117426449, 7.614104473942300, 8.204429526010892, 9.116877787900486, 9.701840751104292, 10.61162418435983, 11.03129783779673, 11.22243794268863, 11.87585623497952, 12.32910848167505, 12.59633584953222, 13.36286584898565, 13.75372841864295, 14.77609115295171, 15.18165707050189, 15.73471084285795
