| L(s) = 1 | + 2-s − 2.69·3-s + 4-s + 2.04·5-s − 2.69·6-s − 3.60·7-s + 8-s + 4.24·9-s + 2.04·10-s + 0.890·11-s − 2.69·12-s − 3.60·14-s − 5.51·15-s + 16-s + 5.96·17-s + 4.24·18-s − 19-s + 2.04·20-s + 9.70·21-s + 0.890·22-s − 8.98·23-s − 2.69·24-s − 0.801·25-s − 3.35·27-s − 3.60·28-s + 5.20·29-s − 5.51·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.916·5-s − 1.09·6-s − 1.36·7-s + 0.353·8-s + 1.41·9-s + 0.647·10-s + 0.268·11-s − 0.777·12-s − 0.963·14-s − 1.42·15-s + 0.250·16-s + 1.44·17-s + 1.00·18-s − 0.229·19-s + 0.458·20-s + 2.11·21-s + 0.189·22-s − 1.87·23-s − 0.549·24-s − 0.160·25-s − 0.646·27-s − 0.681·28-s + 0.967·29-s − 1.00·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.69T + 3T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 0.890T + 11T^{2} \) |
| 17 | \( 1 - 5.96T + 17T^{2} \) |
| 23 | \( 1 + 8.98T + 23T^{2} \) |
| 29 | \( 1 - 5.20T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 9.38T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 4.71T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 7.82T + 61T^{2} \) |
| 67 | \( 1 + 3.13T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 4.80T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 5.42T + 89T^{2} \) |
| 97 | \( 1 + 3.50T + 97T^{2} \) |
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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
−7.27039949504018830065979622648, −6.44524062672604863286262328903, −6.25065758643752297268571257040, −5.59247940900972712899002027594, −5.14654884104770147141748124408, −4.05562267618112172737423640237, −3.41290050389039035738111906997, −2.30347775492518008214648858202, −1.25003049852306150072038090182, 0,
1.25003049852306150072038090182, 2.30347775492518008214648858202, 3.41290050389039035738111906997, 4.05562267618112172737423640237, 5.14654884104770147141748124408, 5.59247940900972712899002027594, 6.25065758643752297268571257040, 6.44524062672604863286262328903, 7.27039949504018830065979622648
