L(s) = 1 | − 3-s + 9-s − 27-s + 2·31-s − 49-s + 2·53-s + 2·79-s + 81-s + 2·83-s − 2·93-s + 2·107-s + ⋯ |
L(s) = 1 | − 3-s + 9-s − 27-s + 2·31-s − 49-s + 2·53-s + 2·79-s + 81-s + 2·83-s − 2·93-s + 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8718242642\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8718242642\) |
\(L(1)\) |
|
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 | \( 1 \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.266379523577035906972890572651, −8.306219165417714900897235426167, −7.53490077743187679786874007706, −6.65591644142831020731948803863, −6.12609166362309969535617682525, −5.18649366365133013363305610145, −4.54169239768385464632995052855, −3.57353442307534142272482538150, −2.28842157838104445595981880507, −0.959214039807738626012217305732,
0.959214039807738626012217305732, 2.28842157838104445595981880507, 3.57353442307534142272482538150, 4.54169239768385464632995052855, 5.18649366365133013363305610145, 6.12609166362309969535617682525, 6.65591644142831020731948803863, 7.53490077743187679786874007706, 8.306219165417714900897235426167, 9.266379523577035906972890572651