L(s) = 1 | − 5-s − 7-s + 9-s − 11-s − 17-s + 19-s + 2·23-s + 35-s − 43-s − 45-s − 47-s + 55-s − 61-s − 63-s − 73-s + 77-s + 81-s + 2·83-s + 85-s − 95-s − 99-s + 2·101-s − 2·115-s + 119-s + ⋯ |
L(s) = 1 | − 5-s − 7-s + 9-s − 11-s − 17-s + 19-s + 2·23-s + 35-s − 43-s − 45-s − 47-s + 55-s − 61-s − 63-s − 73-s + 77-s + 81-s + 2·83-s + 85-s − 95-s − 99-s + 2·101-s − 2·115-s + 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4523541508\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4523541508\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−15.18389011950467638619571036076, −13.35974916352130316794641901241, −12.80383094195036971594468272052, −11.52365494724892801268611564374, −10.37363017330335958179980388288, −9.200011605496345198209119678485, −7.71021732429554568325384670285, −6.75966056696874593751657456611, −4.83345700374897654801370031599, −3.27623368738075539180170835921,
3.27623368738075539180170835921, 4.83345700374897654801370031599, 6.75966056696874593751657456611, 7.71021732429554568325384670285, 9.200011605496345198209119678485, 10.37363017330335958179980388288, 11.52365494724892801268611564374, 12.80383094195036971594468272052, 13.35974916352130316794641901241, 15.18389011950467638619571036076