| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s + 3·11-s − 2·13-s + 16-s + 6·17-s − 2·19-s + 3·20-s − 3·22-s + 6·23-s + 4·25-s + 2·26-s − 9·29-s + 7·31-s − 32-s − 6·34-s − 10·37-s + 2·38-s − 3·40-s − 4·43-s + 3·44-s − 6·46-s + 12·47-s − 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s + 0.904·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.458·19-s + 0.670·20-s − 0.639·22-s + 1.25·23-s + 4/5·25-s + 0.392·26-s − 1.67·29-s + 1.25·31-s − 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.324·38-s − 0.474·40-s − 0.609·43-s + 0.452·44-s − 0.884·46-s + 1.75·47-s − 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536586658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536586658\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−10.02878276265861936276850146227, −9.331076611077021290411627883935, −8.753214920894426804382507545712, −7.56006473562539600008273040402, −6.76340153457321918972152567863, −5.88431208298112457486858305913, −5.10036072121782338740270472262, −3.52994453436524682105467711302, −2.28166780967037125049719495570, −1.22415852747718716380535497945,
1.22415852747718716380535497945, 2.28166780967037125049719495570, 3.52994453436524682105467711302, 5.10036072121782338740270472262, 5.88431208298112457486858305913, 6.76340153457321918972152567863, 7.56006473562539600008273040402, 8.753214920894426804382507545712, 9.331076611077021290411627883935, 10.02878276265861936276850146227
