| L(s) = 1 | + (0.309 + 0.951i)2-s + (0.978 − 0.207i)3-s + (−0.809 + 0.587i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−2.70 − 1.20i)7-s + (−0.809 − 0.587i)8-s + (0.913 − 0.406i)9-s + (0.978 + 0.207i)10-s + (−0.0918 − 0.874i)11-s + (−0.669 + 0.743i)12-s + (−2.33 − 2.59i)13-s + (0.309 − 2.94i)14-s + (0.309 − 0.951i)15-s + (0.309 − 0.951i)16-s + (0.838 − 7.97i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.564 − 0.120i)3-s + (−0.404 + 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−1.02 − 0.455i)7-s + (−0.286 − 0.207i)8-s + (0.304 − 0.135i)9-s + (0.309 + 0.0657i)10-s + (−0.0277 − 0.263i)11-s + (−0.193 + 0.214i)12-s + (−0.648 − 0.719i)13-s + (0.0826 − 0.786i)14-s + (0.0797 − 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.203 − 1.93i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41327 - 0.674527i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41327 - 0.674527i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-3.49 - 4.33i)T \) |
| good | 7 | \( 1 + (2.70 + 1.20i)T + (4.68 + 5.20i)T^{2} \) |
| 11 | \( 1 + (0.0918 + 0.874i)T + (-10.7 + 2.28i)T^{2} \) |
| 13 | \( 1 + (2.33 + 2.59i)T + (-1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.838 + 7.97i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.65 + 1.84i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.80 - 1.30i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.877 - 2.70i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.48 + 4.30i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.87 + 0.822i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-2.45 + 2.73i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-2.05 + 6.31i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (9.06 - 4.03i)T + (35.4 - 39.3i)T^{2} \) |
| 59 | \( 1 + (-3.74 + 0.795i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 + (-3.84 + 6.65i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.88 - 3.95i)T + (47.5 - 52.7i)T^{2} \) |
| 73 | \( 1 + (-1.59 - 15.1i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.930 + 8.84i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-16.4 - 3.49i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (11.8 - 8.58i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.43 + 3.94i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.637100557267157457987479127685, −9.176823969071795338653072166940, −8.199362291013668480619433387034, −7.19423856426922722954627664503, −6.84326769490709858662976326373, −5.50537780806011291434904250008, −4.83636019286857061068589029508, −3.47890161607637840551172476292, −2.74772576956250244179275686165, −0.63421532934829226326397563276,
1.78161638987632424243058073296, 2.78432406374001228027563933847, 3.65312309980390254485721774058, 4.62511288716071719874761458481, 5.95390222208248967260835454926, 6.58614705357301710365973539163, 7.81230845205530737560606062086, 8.715924479269189954219325228543, 9.677100403791677362833407325667, 9.965546464233385989213988308818
