L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 4·11-s + 2·13-s − 15-s − 6·17-s − 19-s + 4·21-s − 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s − 4·33-s − 4·35-s + 10·37-s − 2·39-s + 2·41-s + 45-s + 4·47-s + 9·49-s + 6·51-s + 2·53-s + 4·55-s + 57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s − 1.45·17-s − 0.229·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s + 0.312·41-s + 0.149·45-s + 0.583·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.539·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.247622961\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247622961\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.597868213785013283294424214525, −7.29727168098159772066066874094, −6.67346952173222818736818531256, −6.17701843493987983428915779478, −5.73811103545833013445606483311, −4.40522243267350251888189588326, −3.94543152310830867263732506887, −2.91353064096016451096410866873, −1.88609460132414588878093443132, −0.63292907005131187915961561972,
0.63292907005131187915961561972, 1.88609460132414588878093443132, 2.91353064096016451096410866873, 3.94543152310830867263732506887, 4.40522243267350251888189588326, 5.73811103545833013445606483311, 6.17701843493987983428915779478, 6.67346952173222818736818531256, 7.29727168098159772066066874094, 8.597868213785013283294424214525