L(s) = 1 | − 5-s + 9-s + 2·13-s + 25-s − 2·37-s − 2·41-s − 45-s − 49-s + 2·53-s − 2·65-s + 81-s − 2·89-s + 2·117-s + ⋯ |
L(s) = 1 | − 5-s + 9-s + 2·13-s + 25-s − 2·37-s − 2·41-s − 45-s − 49-s + 2·53-s − 2·65-s + 81-s − 2·89-s + 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8943906129\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8943906129\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−10.79495837168484138612076670698, −10.09874468729156596722724484143, −8.785888650409807130869408415447, −8.310120984203962583004789925280, −7.19903958631570605268935541489, −6.51778033020758742396704834596, −5.21270997532034263218530541523, −4.04457968765298506877627190712, −3.41835731076545275239064437262, −1.45578699018930483007679514468,
1.45578699018930483007679514468, 3.41835731076545275239064437262, 4.04457968765298506877627190712, 5.21270997532034263218530541523, 6.51778033020758742396704834596, 7.19903958631570605268935541489, 8.310120984203962583004789925280, 8.785888650409807130869408415447, 10.09874468729156596722724484143, 10.79495837168484138612076670698