| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.005979036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.005979036\) |
| \(L(\frac{3}{2})\) |
|
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
−7.79135202024277334389639114403, −7.15922049480210701139319720327, −6.32854729307507255571323398252, −5.86405387253474759787898954785, −4.64101075926790658806218447437, −4.07494342133225173372309331792, −3.50933640798374715298025226500, −2.86472456216643268078313135158, −1.51892061640086881884189679623, −0.909468114889031403038423095373,
0.909468114889031403038423095373, 1.51892061640086881884189679623, 2.86472456216643268078313135158, 3.50933640798374715298025226500, 4.07494342133225173372309331792, 4.64101075926790658806218447437, 5.86405387253474759787898954785, 6.32854729307507255571323398252, 7.15922049480210701139319720327, 7.79135202024277334389639114403
