| L(s) = 1 | + 3.32·3-s − 5-s + 5.02·7-s + 8.02·9-s + 1.70·11-s − 13-s − 3.32·15-s − 4.64·17-s − 4.34·19-s + 16.6·21-s + 0.679·23-s + 25-s + 16.6·27-s + 1.02·29-s + 2.29·31-s + 5.67·33-s − 5.02·35-s + 1.61·37-s − 3.32·39-s + 4.64·41-s − 3.32·43-s − 8.02·45-s − 1.02·47-s + 18.2·49-s − 15.4·51-s + 9.41·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 0.447·5-s + 1.90·7-s + 2.67·9-s + 0.514·11-s − 0.277·13-s − 0.857·15-s − 1.12·17-s − 0.997·19-s + 3.64·21-s + 0.141·23-s + 0.200·25-s + 3.21·27-s + 0.190·29-s + 0.411·31-s + 0.987·33-s − 0.849·35-s + 0.265·37-s − 0.531·39-s + 0.724·41-s − 0.506·43-s − 1.19·45-s − 0.149·47-s + 2.61·49-s − 2.15·51-s + 1.29·53-s − 0.230·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.797625771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.797625771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 7 | \( 1 - 5.02T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 0.679T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 2.29T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 + 3.32T + 43T^{2} \) |
| 47 | \( 1 + 1.02T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 + 8.93T + 59T^{2} \) |
| 61 | \( 1 + 9.02T + 61T^{2} \) |
| 67 | \( 1 - 5.61T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 + 8.25T + 83T^{2} \) |
| 89 | \( 1 - 1.22T + 89T^{2} \) |
| 97 | \( 1 + 0.0565T + 97T^{2} \) |
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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
−8.272049689416930894781327447053, −8.022521741411066927972931592415, −7.24713879733767510954848673185, −6.56005578403982917384905098615, −5.05868520371170559248896924441, −4.28867038369510032566603127732, −4.03648398860357021755279822167, −2.74965397328258641501894054694, −2.09536299025809941333889364386, −1.30837426207716274057522070298,
1.30837426207716274057522070298, 2.09536299025809941333889364386, 2.74965397328258641501894054694, 4.03648398860357021755279822167, 4.28867038369510032566603127732, 5.05868520371170559248896924441, 6.56005578403982917384905098615, 7.24713879733767510954848673185, 8.022521741411066927972931592415, 8.272049689416930894781327447053
