| L(s) = 1 | + 1.65·3-s − 5-s + 4.70·7-s − 0.259·9-s − 5.05·13-s − 1.65·15-s − 5.31·17-s + 2.25·19-s + 7.79·21-s − 1.05·23-s + 25-s − 5.39·27-s − 2.79·29-s − 3.74·31-s − 4.70·35-s − 0.791·37-s − 8.36·39-s + 6.15·41-s − 2.70·43-s + 0.259·45-s + 11.2·47-s + 15.1·49-s − 8.79·51-s + 1.05·53-s + 3.74·57-s − 4.53·59-s − 9.88·61-s + ⋯ |
| L(s) = 1 | + 0.955·3-s − 0.447·5-s + 1.77·7-s − 0.0865·9-s − 1.40·13-s − 0.427·15-s − 1.28·17-s + 0.518·19-s + 1.70·21-s − 0.219·23-s + 0.200·25-s − 1.03·27-s − 0.518·29-s − 0.671·31-s − 0.795·35-s − 0.130·37-s − 1.33·39-s + 0.961·41-s − 0.412·43-s + 0.0386·45-s + 1.63·47-s + 2.16·49-s − 1.23·51-s + 0.144·53-s + 0.495·57-s − 0.590·59-s − 1.26·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.65T + 3T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 13 | \( 1 + 5.05T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 23 | \( 1 + 1.05T + 23T^{2} \) |
| 29 | \( 1 + 2.79T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 + 0.791T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.05T + 53T^{2} \) |
| 59 | \( 1 + 4.53T + 59T^{2} \) |
| 61 | \( 1 + 9.88T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 1.05T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 9.41T + 79T^{2} \) |
| 83 | \( 1 + 3.15T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.53093515513403220607340576732, −7.06016786566419472800764430180, −5.81191121823784548861856247547, −5.14893901268951749441661851069, −4.45597836300283903773649081818, −3.95630575912247463450561928451, −2.80265928830987154467321892739, −2.27885718127817844885052804388, −1.49574850397532985872836933486, 0,
1.49574850397532985872836933486, 2.27885718127817844885052804388, 2.80265928830987154467321892739, 3.95630575912247463450561928451, 4.45597836300283903773649081818, 5.14893901268951749441661851069, 5.81191121823784548861856247547, 7.06016786566419472800764430180, 7.53093515513403220607340576732
