| L(s) = 1 | + 3-s + 3.52·5-s + 3.24·7-s + 9-s + 11-s + 13-s + 3.52·15-s − 5.48·17-s + 2.44·19-s + 3.24·21-s − 2.29·23-s + 7.45·25-s + 27-s + 9.32·29-s − 2.32·31-s + 33-s + 11.4·35-s − 11.4·37-s + 39-s − 10.0·41-s + 7.09·43-s + 3.52·45-s + 2.48·47-s + 3.55·49-s − 5.48·51-s + 13.6·53-s + 3.52·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.57·5-s + 1.22·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s + 0.911·15-s − 1.33·17-s + 0.561·19-s + 0.708·21-s − 0.478·23-s + 1.49·25-s + 0.192·27-s + 1.73·29-s − 0.418·31-s + 0.174·33-s + 1.93·35-s − 1.88·37-s + 0.160·39-s − 1.57·41-s + 1.08·43-s + 0.526·45-s + 0.362·47-s + 0.507·49-s − 0.768·51-s + 1.87·53-s + 0.475·55-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
| \(L(1)\) |
\(\approx\) |
\(4.394492410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.394492410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.52T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 5.35T + 59T^{2} \) |
| 61 | \( 1 + 7.71T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.76T + 71T^{2} \) |
| 73 | \( 1 - 1.80T + 73T^{2} \) |
| 79 | \( 1 + 3.26T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 2.06T + 89T^{2} \) |
| 97 | \( 1 + 5.04T + 97T^{2} \) |
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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
−8.225325549764904033650524953708, −7.11949879599935645799159087250, −6.66458309020129554889995246029, −5.79279225924120180045012719339, −5.12018757089083878180399294952, −4.51128744757934121803780445878, −3.52760204514209158790318238752, −2.39906283848335960922679132612, −1.95106675103046680636330258061, −1.14364279698951483970563004717,
1.14364279698951483970563004717, 1.95106675103046680636330258061, 2.39906283848335960922679132612, 3.52760204514209158790318238752, 4.51128744757934121803780445878, 5.12018757089083878180399294952, 5.79279225924120180045012719339, 6.66458309020129554889995246029, 7.11949879599935645799159087250, 8.225325549764904033650524953708
