| L(s) = 1 | − 16.6·7-s + 19.1·11-s − 61.7·13-s − 30.3·17-s + 59.1·19-s + 205.·23-s − 8.38·29-s + 331.·31-s − 266.·37-s + 320.·41-s − 83.1·43-s − 276.·47-s − 64.2·49-s + 390.·53-s − 779.·59-s − 483.·61-s − 123.·67-s − 187.·71-s + 778.·73-s − 320.·77-s − 446.·79-s − 1.05e3·83-s + 94.8·89-s + 1.03e3·91-s − 252.·97-s − 37.9·101-s − 94.4·103-s + ⋯ |
| L(s) = 1 | − 0.901·7-s + 0.526·11-s − 1.31·13-s − 0.433·17-s + 0.714·19-s + 1.86·23-s − 0.0536·29-s + 1.91·31-s − 1.18·37-s + 1.22·41-s − 0.294·43-s − 0.857·47-s − 0.187·49-s + 1.01·53-s − 1.71·59-s − 1.01·61-s − 0.225·67-s − 0.312·71-s + 1.24·73-s − 0.474·77-s − 0.635·79-s − 1.39·83-s + 0.112·89-s + 1.18·91-s − 0.263·97-s − 0.0373·101-s − 0.0903·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 16.6T + 343T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 30.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 8.38T + 2.43e4T^{2} \) |
| 31 | \( 1 - 331.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 83.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 390.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 779.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 483.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 187.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 446.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 94.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 252.T + 9.12e5T^{2} \) |
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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
−8.677373092284543392796999438480, −7.57544154371134903254091114543, −6.91341254025872192176530518118, −6.27950723565346105495161985516, −5.16221185233703709227695750560, −4.47743669349494337053145682293, −3.25113615639957590548839739909, −2.62960124638855307720602196103, −1.19150391239609126130263601947, 0,
1.19150391239609126130263601947, 2.62960124638855307720602196103, 3.25113615639957590548839739909, 4.47743669349494337053145682293, 5.16221185233703709227695750560, 6.27950723565346105495161985516, 6.91341254025872192176530518118, 7.57544154371134903254091114543, 8.677373092284543392796999438480
