| L(s) = 1 | − 3·3-s − 19.4·5-s − 20.3·7-s + 9·9-s − 8.19·13-s + 58.4·15-s + 92.0·17-s − 51.8·19-s + 60.9·21-s − 149.·23-s + 254.·25-s − 27·27-s + 171.·29-s + 273.·31-s + 395.·35-s + 169.·37-s + 24.5·39-s − 227.·41-s − 338.·43-s − 175.·45-s + 166.·47-s + 69.6·49-s − 276.·51-s + 314.·53-s + 155.·57-s + 580.·59-s + 586.·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.74·5-s − 1.09·7-s + 0.333·9-s − 0.174·13-s + 1.00·15-s + 1.31·17-s − 0.626·19-s + 0.633·21-s − 1.35·23-s + 2.03·25-s − 0.192·27-s + 1.09·29-s + 1.58·31-s + 1.91·35-s + 0.753·37-s + 0.100·39-s − 0.865·41-s − 1.20·43-s − 0.580·45-s + 0.516·47-s + 0.203·49-s − 0.757·51-s + 0.814·53-s + 0.361·57-s + 1.28·59-s + 1.23·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 19.4T + 125T^{2} \) |
| 7 | \( 1 + 20.3T + 343T^{2} \) |
| 13 | \( 1 + 8.19T + 2.19e3T^{2} \) |
| 17 | \( 1 - 92.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 51.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 338.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 166.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 314.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 580.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 586.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 392.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 934.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.49e3T + 9.12e5T^{2} \) |
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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
−8.390486629363226787017256130663, −8.092681007490855574632836755870, −7.00626573452921267905776349765, −6.48803299590675840033911820879, −5.41310992767672923162720352723, −4.32457660522063837804778912396, −3.71096227676734647064183459284, −2.77297008930315133108831638528, −0.876758276788321100955405356963, 0,
0.876758276788321100955405356963, 2.77297008930315133108831638528, 3.71096227676734647064183459284, 4.32457660522063837804778912396, 5.41310992767672923162720352723, 6.48803299590675840033911820879, 7.00626573452921267905776349765, 8.092681007490855574632836755870, 8.390486629363226787017256130663
