| 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
−9.965546464233385989213988308818, −9.677100403791677362833407325667, −8.715924479269189954219325228543, −7.81230845205530737560606062086, −6.58614705357301710365973539163, −5.95390222208248967260835454926, −4.62511288716071719874761458481, −3.65312309980390254485721774058, −2.78432406374001228027563933847, −1.78161638987632424243058073296,
0.63421532934829226326397563276, 2.74772576956250244179275686165, 3.47890161607637840551172476292, 4.83636019286857061068589029508, 5.50537780806011291434904250008, 6.84326769490709858662976326373, 7.19423856426922722954627664503, 8.199362291013668480619433387034, 9.176823969071795338653072166940, 9.637100557267157457987479127685
