L(s) = 1 | − 3-s + 2·5-s + 9-s − 6·11-s − 3·13-s − 2·15-s + 4·17-s − 5·19-s − 4·23-s − 25-s − 27-s − 4·29-s + 7·31-s + 6·33-s − 9·37-s + 3·39-s − 2·41-s − 43-s + 2·45-s + 2·47-s − 4·51-s + 8·53-s − 12·55-s + 5·57-s + 10·61-s − 6·65-s − 15·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.80·11-s − 0.832·13-s − 0.516·15-s + 0.970·17-s − 1.14·19-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.742·29-s + 1.25·31-s + 1.04·33-s − 1.47·37-s + 0.480·39-s − 0.312·41-s − 0.152·43-s + 0.298·45-s + 0.291·47-s − 0.560·51-s + 1.09·53-s − 1.61·55-s + 0.662·57-s + 1.28·61-s − 0.744·65-s − 1.83·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 15 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.681721320131094163215273255189, −8.460071520116083697903537739907, −7.70587249118750730592766123774, −6.81232371947482166887069150763, −5.69453282174017204887014619372, −5.39310938366525563858610112111, −4.31058491195255984662115576742, −2.81439508368098570027009824752, −1.88941033513530829976904950599, 0,
1.88941033513530829976904950599, 2.81439508368098570027009824752, 4.31058491195255984662115576742, 5.39310938366525563858610112111, 5.69453282174017204887014619372, 6.81232371947482166887069150763, 7.70587249118750730592766123774, 8.460071520116083697903537739907, 9.681721320131094163215273255189