| L(s) = 1 | + 3-s + 4·5-s − 7-s + 9-s − 11-s + 6·13-s + 4·15-s + 2·19-s − 21-s − 4·23-s + 11·25-s + 27-s − 2·29-s + 2·31-s − 33-s − 4·35-s + 2·37-s + 6·39-s + 4·43-s + 4·45-s − 6·47-s + 49-s + 2·53-s − 4·55-s + 2·57-s + 14·61-s − 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s + 1.03·15-s + 0.458·19-s − 0.218·21-s − 0.834·23-s + 11/5·25-s + 0.192·27-s − 0.371·29-s + 0.359·31-s − 0.174·33-s − 0.676·35-s + 0.328·37-s + 0.960·39-s + 0.609·43-s + 0.596·45-s − 0.875·47-s + 1/7·49-s + 0.274·53-s − 0.539·55-s + 0.264·57-s + 1.79·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.039591783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.039591783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.089845432955761391898877334947, −7.07226117189654732272179235239, −6.39730068704497360053429151819, −5.84922001149124764426400687445, −5.33384232680319303215146871320, −4.23853713335788444549584454970, −3.40524636477070840793682383835, −2.61840495749355237814677161610, −1.85643496417184993589238337200, −1.05708410431101876420219167418,
1.05708410431101876420219167418, 1.85643496417184993589238337200, 2.61840495749355237814677161610, 3.40524636477070840793682383835, 4.23853713335788444549584454970, 5.33384232680319303215146871320, 5.84922001149124764426400687445, 6.39730068704497360053429151819, 7.07226117189654732272179235239, 8.089845432955761391898877334947
