L(s) = 1 | − 3-s + 5-s + 3·7-s + 9-s + 3·11-s + 13-s − 15-s − 3·17-s + 6·19-s − 3·21-s + 4·23-s + 25-s − 27-s + 29-s + 4·31-s − 3·33-s + 3·35-s − 4·37-s − 39-s + 2·41-s − 4·43-s + 45-s + 3·47-s + 2·49-s + 3·51-s + 6·53-s + 3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s + 1.37·19-s − 0.654·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.185·29-s + 0.718·31-s − 0.522·33-s + 0.507·35-s − 0.657·37-s − 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.149·45-s + 0.437·47-s + 2/7·49-s + 0.420·51-s + 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.557245328\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.557245328\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.017374250163769451508703613832, −6.97178977422643903235387332619, −6.74121786877230458825582625580, −5.67746442840046865684301129584, −5.20885456047408429545513886600, −4.50152080957432099664700124918, −3.72010592995954142487840233743, −2.61682242472160035374681962520, −1.57747335906223423894010410579, −0.939047166383907160767671412639,
0.939047166383907160767671412639, 1.57747335906223423894010410579, 2.61682242472160035374681962520, 3.72010592995954142487840233743, 4.50152080957432099664700124918, 5.20885456047408429545513886600, 5.67746442840046865684301129584, 6.74121786877230458825582625580, 6.97178977422643903235387332619, 8.017374250163769451508703613832