| L(s) = 1 | + 3-s + 3.70·7-s + 9-s − 5.70·11-s + 13-s − 5.70·17-s − 2·19-s + 3.70·21-s − 0.298·23-s + 27-s − 3.40·29-s + 3.40·31-s − 5.70·33-s + 0.298·37-s + 39-s + 4.29·41-s − 4·43-s − 11.4·47-s + 6.70·49-s − 5.70·51-s − 4.29·53-s − 2·57-s + 10.8·59-s − 3.70·61-s + 3.70·63-s + 4·67-s − 0.298·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.39·7-s + 0.333·9-s − 1.71·11-s + 0.277·13-s − 1.38·17-s − 0.458·19-s + 0.807·21-s − 0.0622·23-s + 0.192·27-s − 0.631·29-s + 0.611·31-s − 0.992·33-s + 0.0490·37-s + 0.160·39-s + 0.671·41-s − 0.609·43-s − 1.66·47-s + 0.957·49-s − 0.798·51-s − 0.590·53-s − 0.264·57-s + 1.40·59-s − 0.473·61-s + 0.466·63-s + 0.488·67-s − 0.0359·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 + 5.70T + 11T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 0.298T + 23T^{2} \) |
| 29 | \( 1 + 3.40T + 29T^{2} \) |
| 31 | \( 1 - 3.40T + 31T^{2} \) |
| 37 | \( 1 - 0.298T + 37T^{2} \) |
| 41 | \( 1 - 4.29T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.29T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 0.596T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 1.10T + 97T^{2} \) |
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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
−7.67785063904964523830487751811, −7.00720693043923507696612725962, −6.10187293409555604159188222360, −5.20913974332270176626736752648, −4.71053643156127050637246655074, −4.04825501854536297769230575033, −2.90866040352754123176957515802, −2.25772694655059107006728264401, −1.52289531914570518966651386584, 0,
1.52289531914570518966651386584, 2.25772694655059107006728264401, 2.90866040352754123176957515802, 4.04825501854536297769230575033, 4.71053643156127050637246655074, 5.20913974332270176626736752648, 6.10187293409555604159188222360, 7.00720693043923507696612725962, 7.67785063904964523830487751811
