| L(s) = 1 | + 3-s + 4·7-s + 9-s − 6·11-s − 13-s − 17-s − 2·19-s + 4·21-s + 4·23-s + 27-s + 8·29-s + 31-s − 6·33-s − 37-s − 39-s + 2·41-s − 8·43-s + 47-s + 9·49-s − 51-s + 7·53-s − 2·57-s − 13·59-s + 6·61-s + 4·63-s − 5·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.242·17-s − 0.458·19-s + 0.872·21-s + 0.834·23-s + 0.192·27-s + 1.48·29-s + 0.179·31-s − 1.04·33-s − 0.164·37-s − 0.160·39-s + 0.312·41-s − 1.21·43-s + 0.145·47-s + 9/7·49-s − 0.140·51-s + 0.961·53-s − 0.264·57-s − 1.69·59-s + 0.768·61-s + 0.503·63-s − 0.610·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.53810842220662, −13.03600524437162, −12.49516658339023, −12.00393288543399, −11.55394583811616, −10.83022418674892, −10.64620693900028, −10.26799266149506, −9.602927300516974, −9.012029468917086, −8.378810604927775, −8.246448092818745, −7.799510135460481, −7.236416439004366, −6.813092162407905, −6.048598959600467, −5.335691977029257, −5.023597374449471, −4.570133573866456, −4.120444508724873, −3.132222136683736, −2.762288048011936, −2.228333175576170, −1.637771335011260, −0.9302585564197157, 0,
0.9302585564197157, 1.637771335011260, 2.228333175576170, 2.762288048011936, 3.132222136683736, 4.120444508724873, 4.570133573866456, 5.023597374449471, 5.335691977029257, 6.048598959600467, 6.813092162407905, 7.236416439004366, 7.799510135460481, 8.246448092818745, 8.378810604927775, 9.012029468917086, 9.602927300516974, 10.26799266149506, 10.64620693900028, 10.83022418674892, 11.55394583811616, 12.00393288543399, 12.49516658339023, 13.03600524437162, 13.53810842220662
