| L(s) = 1 | + 4·3-s + 20·5-s + 36·7-s + 8·9-s − 66·13-s + 80·15-s + 134·17-s + 232·19-s + 144·21-s − 220·23-s + 275·25-s + 108·27-s + 720·35-s + 130·37-s − 264·39-s − 608·41-s + 308·43-s + 160·45-s − 612·47-s + 648·49-s + 536·51-s + 434·53-s + 928·57-s − 408·59-s − 1.49e3·61-s + 288·63-s − 1.32e3·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 1.78·5-s + 1.94·7-s + 8/27·9-s − 1.40·13-s + 1.37·15-s + 1.91·17-s + 2.80·19-s + 1.49·21-s − 1.99·23-s + 11/5·25-s + 0.769·27-s + 3.47·35-s + 0.577·37-s − 1.08·39-s − 2.31·41-s + 1.09·43-s + 0.530·45-s − 1.89·47-s + 1.88·49-s + 1.47·51-s + 1.12·53-s + 2.15·57-s − 0.900·59-s − 3.14·61-s + 0.575·63-s − 2.51·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.811165723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.811165723\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2406 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 134 T + 8978 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 220 T + 24200 T^{2} + 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31354 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 54958 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 130 T + 8450 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 304 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 308 T + 47432 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 612 T + 187272 T^{2} + 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 434 T + 94178 T^{2} - 434 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 204 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 748 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 332 T + 55112 T^{2} - 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 441246 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 554 T + 153458 T^{2} - 554 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1232 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 580 T + 168200 T^{2} + 580 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 769938 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1302 T + 847602 T^{2} + 1302 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41577583371779528578820677119, −12.33688456545879801100311109220, −11.70634966935305238307471963785, −11.30547478032663577973804810579, −10.22988935867973660869359616584, −10.16128774201427018710251342530, −9.620034496920402856237095567649, −9.373511146577893307420543583198, −8.330261778820183232615039516091, −8.154134713247679709817563155567, −7.43976814870619673582806182366, −7.18546340490091417075565607155, −5.96748186610131120196303269702, −5.52252972511853437216764367996, −5.08349480767729650472206831513, −4.54537238748173856770450236651, −3.22206768094653909460068031391, −2.68842716885719850107681810729, −1.58781607381382716610943235706, −1.43082213576717280622176868515,
1.43082213576717280622176868515, 1.58781607381382716610943235706, 2.68842716885719850107681810729, 3.22206768094653909460068031391, 4.54537238748173856770450236651, 5.08349480767729650472206831513, 5.52252972511853437216764367996, 5.96748186610131120196303269702, 7.18546340490091417075565607155, 7.43976814870619673582806182366, 8.154134713247679709817563155567, 8.330261778820183232615039516091, 9.373511146577893307420543583198, 9.620034496920402856237095567649, 10.16128774201427018710251342530, 10.22988935867973660869359616584, 11.30547478032663577973804810579, 11.70634966935305238307471963785, 12.33688456545879801100311109220, 12.41577583371779528578820677119
