| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s − 50.8·11-s − 16.4·13-s + 25.9·17-s + 106.·19-s − 21·21-s − 42.5·23-s − 27·27-s + 64.5·29-s − 40.4·31-s + 152.·33-s + 82.7·37-s + 49.4·39-s + 80.2·41-s + 184.·43-s − 115.·47-s + 49·49-s − 77.7·51-s − 643.·53-s − 318.·57-s + 526.·59-s − 702.·61-s + 63·63-s + 832.·67-s + 127.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 0.333·9-s − 1.39·11-s − 0.351·13-s + 0.369·17-s + 1.28·19-s − 0.218·21-s − 0.385·23-s − 0.192·27-s + 0.413·29-s − 0.234·31-s + 0.804·33-s + 0.367·37-s + 0.203·39-s + 0.305·41-s + 0.655·43-s − 0.357·47-s + 0.142·49-s − 0.213·51-s − 1.66·53-s − 0.739·57-s + 1.16·59-s − 1.47·61-s + 0.125·63-s + 1.51·67-s + 0.222·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 + 50.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 80.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 115.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 643.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 702.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 832.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 487.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 143.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.26e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 860.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−8.045054680979144882343468328625, −7.73728547558054443798302536438, −6.83452819827806847635656694746, −5.78592574452618892769293564433, −5.22769962613234369295375236007, −4.51501322180475690878346137073, −3.30027839810003056071937738987, −2.35175019827560525961377158775, −1.12574185109972005041975757413, 0,
1.12574185109972005041975757413, 2.35175019827560525961377158775, 3.30027839810003056071937738987, 4.51501322180475690878346137073, 5.22769962613234369295375236007, 5.78592574452618892769293564433, 6.83452819827806847635656694746, 7.73728547558054443798302536438, 8.045054680979144882343468328625
