| L(s) = 1 | + 2.14·2-s − 3-s + 2.60·4-s − 1.91·5-s − 2.14·6-s + 7-s + 1.29·8-s + 9-s − 4.11·10-s + 1.42·11-s − 2.60·12-s + 2.14·14-s + 1.91·15-s − 2.42·16-s − 6.47·17-s + 2.14·18-s + 5.74·19-s − 4.99·20-s − 21-s + 3.05·22-s − 2.14·23-s − 1.29·24-s − 1.31·25-s − 27-s + 2.60·28-s − 7.63·29-s + 4.11·30-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 0.577·3-s + 1.30·4-s − 0.858·5-s − 0.876·6-s + 0.377·7-s + 0.458·8-s + 0.333·9-s − 1.30·10-s + 0.428·11-s − 0.751·12-s + 0.573·14-s + 0.495·15-s − 0.606·16-s − 1.56·17-s + 0.505·18-s + 1.31·19-s − 1.11·20-s − 0.218·21-s + 0.650·22-s − 0.446·23-s − 0.264·24-s − 0.263·25-s − 0.192·27-s + 0.492·28-s − 1.41·29-s + 0.752·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 1.91T + 5T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 - 5.74T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 - 7.82T + 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 + 0.362T + 47T^{2} \) |
| 53 | \( 1 - 0.524T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 + 0.405T + 67T^{2} \) |
| 71 | \( 1 + 0.243T + 71T^{2} \) |
| 73 | \( 1 - 0.330T + 73T^{2} \) |
| 79 | \( 1 - 8.64T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + 17.1T + 97T^{2} \) |
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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
−7.85605638631311318330955315281, −7.25129079896188633693450707946, −6.49441145181362792828346493865, −5.75202086887264667500198807028, −5.07796288179995332077821759460, −4.23388113374940193054668793219, −3.90792978439098461320793100286, −2.85727008364491258700532343089, −1.70057099959222982952110966948, 0,
1.70057099959222982952110966948, 2.85727008364491258700532343089, 3.90792978439098461320793100286, 4.23388113374940193054668793219, 5.07796288179995332077821759460, 5.75202086887264667500198807028, 6.49441145181362792828346493865, 7.25129079896188633693450707946, 7.85605638631311318330955315281
