| L(s) = 1 | − 3.02·3-s − 1.51·5-s − 0.303·7-s + 6.17·9-s + 11-s − 13-s + 4.57·15-s + 2.62·17-s − 5.11·19-s + 0.919·21-s + 0.820·23-s − 2.71·25-s − 9.62·27-s − 1.69·29-s + 3.60·31-s − 3.02·33-s + 0.458·35-s + 5.23·37-s + 3.02·39-s + 3.95·41-s − 4.25·43-s − 9.33·45-s − 1.93·47-s − 6.90·49-s − 7.96·51-s − 2.09·53-s − 1.51·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.675·5-s − 0.114·7-s + 2.05·9-s + 0.301·11-s − 0.277·13-s + 1.18·15-s + 0.637·17-s − 1.17·19-s + 0.200·21-s + 0.171·23-s − 0.543·25-s − 1.85·27-s − 0.313·29-s + 0.646·31-s − 0.527·33-s + 0.0775·35-s + 0.861·37-s + 0.485·39-s + 0.617·41-s − 0.648·43-s − 1.39·45-s − 0.282·47-s − 0.986·49-s − 1.11·51-s − 0.288·53-s − 0.203·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.02T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 0.303T + 7T^{2} \) |
| 17 | \( 1 - 2.62T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 0.820T + 23T^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 5.23T + 37T^{2} \) |
| 41 | \( 1 - 3.95T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 + 1.93T + 47T^{2} \) |
| 53 | \( 1 + 2.09T + 53T^{2} \) |
| 59 | \( 1 - 5.59T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 3.98T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 7.38T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.18166717697053353290391910601, −6.62100883089676310985272398661, −5.98136239638166967165272405682, −5.44449884385218031960702402617, −4.52746188597779923970827613434, −4.21604930245990136325526601754, −3.21859439233190974058336024050, −1.94117128850624179756848199555, −0.880293682437463385900769937205, 0,
0.880293682437463385900769937205, 1.94117128850624179756848199555, 3.21859439233190974058336024050, 4.21604930245990136325526601754, 4.52746188597779923970827613434, 5.44449884385218031960702402617, 5.98136239638166967165272405682, 6.62100883089676310985272398661, 7.18166717697053353290391910601
