| L(s) = 1 | + 3-s + 0.699·5-s − 3.79·7-s + 9-s − 11-s − 13-s + 0.699·15-s − 5.09·17-s + 0.538·19-s − 3.79·21-s + 7.18·23-s − 4.51·25-s + 27-s − 9.14·29-s − 1.95·31-s − 33-s − 2.65·35-s − 1.39·37-s − 39-s − 1.79·41-s + 10.8·43-s + 0.699·45-s + 13.5·47-s + 7.37·49-s − 5.09·51-s + 9.90·53-s − 0.699·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.312·5-s − 1.43·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s + 0.180·15-s − 1.23·17-s + 0.123·19-s − 0.827·21-s + 1.49·23-s − 0.902·25-s + 0.192·27-s − 1.69·29-s − 0.350·31-s − 0.174·33-s − 0.448·35-s − 0.229·37-s − 0.160·39-s − 0.279·41-s + 1.65·43-s + 0.104·45-s + 1.98·47-s + 1.05·49-s − 0.712·51-s + 1.36·53-s − 0.0942·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.727956506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.727956506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.699T + 5T^{2} \) |
| 7 | \( 1 + 3.79T + 7T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 - 0.538T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 + 9.14T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 + 1.79T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 + 5.72T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 - 9.90T + 71T^{2} \) |
| 73 | \( 1 - 3.83T + 73T^{2} \) |
| 79 | \( 1 + 1.37T + 79T^{2} \) |
| 83 | \( 1 - 4.32T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 - 1.95T + 97T^{2} \) |
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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
−7.911814886753078920583876993634, −7.10075715077162842875830829203, −6.79989588799441857040848321625, −5.82660136732763161536209615136, −5.27289291138736477083168630756, −4.10335085241219726420427450509, −3.60375309521087566139096355299, −2.62385752105451812353665252685, −2.13678824245385146321483853367, −0.62022020260467284746473901043,
0.62022020260467284746473901043, 2.13678824245385146321483853367, 2.62385752105451812353665252685, 3.60375309521087566139096355299, 4.10335085241219726420427450509, 5.27289291138736477083168630756, 5.82660136732763161536209615136, 6.79989588799441857040848321625, 7.10075715077162842875830829203, 7.911814886753078920583876993634
