| L(s) = 1 | + (−0.777 − 0.629i)2-s + (−0.964 + 1.48i)3-s + (0.207 + 0.978i)4-s + (−2.18 − 0.487i)5-s + (1.68 − 0.547i)6-s + (1.85 + 1.88i)7-s + (0.453 − 0.891i)8-s + (−0.0557 − 0.125i)9-s + (1.38 + 1.75i)10-s + (−4.94 − 2.20i)11-s + (−1.65 − 0.634i)12-s + (3.81 − 0.604i)13-s + (−0.259 − 2.63i)14-s + (2.82 − 2.77i)15-s + (−0.913 + 0.406i)16-s + (−5.49 + 0.287i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 − 0.444i)2-s + (−0.556 + 0.857i)3-s + (0.103 + 0.489i)4-s + (−0.975 − 0.217i)5-s + (0.687 − 0.223i)6-s + (0.702 + 0.711i)7-s + (0.160 − 0.315i)8-s + (−0.0185 − 0.0417i)9-s + (0.439 + 0.554i)10-s + (−1.49 − 0.664i)11-s + (−0.477 − 0.183i)12-s + (1.05 − 0.167i)13-s + (−0.0694 − 0.703i)14-s + (0.730 − 0.715i)15-s + (−0.228 + 0.101i)16-s + (−1.33 + 0.0698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00578127 - 0.0634627i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00578127 - 0.0634627i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.777 + 0.629i)T \) |
| 5 | \( 1 + (2.18 + 0.487i)T \) |
| 7 | \( 1 + (-1.85 - 1.88i)T \) |
| good | 3 | \( 1 + (0.964 - 1.48i)T + (-1.22 - 2.74i)T^{2} \) |
| 11 | \( 1 + (4.94 + 2.20i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-3.81 + 0.604i)T + (12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (5.49 - 0.287i)T + (16.9 - 1.77i)T^{2} \) |
| 19 | \( 1 + (7.36 + 1.56i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-2.16 + 2.67i)T + (-4.78 - 22.4i)T^{2} \) |
| 29 | \( 1 + (4.18 + 1.35i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.64 + 4.18i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.06 + 2.78i)T + (-27.4 - 24.7i)T^{2} \) |
| 41 | \( 1 + (0.101 - 0.139i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (3.80 + 3.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.706 - 13.4i)T + (-46.7 - 4.91i)T^{2} \) |
| 53 | \( 1 + (-6.79 - 4.41i)T + (21.5 + 48.4i)T^{2} \) |
| 59 | \( 1 + (0.378 - 3.60i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-0.292 + 0.0307i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (0.169 + 3.24i)T + (-66.6 + 7.00i)T^{2} \) |
| 71 | \( 1 + (-0.401 + 1.23i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (6.71 - 2.57i)T + (54.2 - 48.8i)T^{2} \) |
| 79 | \( 1 + (-1.05 + 0.949i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-3.76 - 1.91i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.208 - 1.98i)T + (-87.0 + 18.5i)T^{2} \) |
| 97 | \( 1 + (-7.02 + 3.57i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.43268383727879394569884445530, −10.88381383062616781067420348836, −10.69941757244503777847457437665, −9.019118600923156608013579340678, −8.475194121663862799182687118013, −7.63462469489044118744999751195, −5.99858726311636600331537258530, −4.85007622167498004987990515057, −4.00968326265060691299413155121, −2.40901260359059905051293176816,
0.05409609429672631041572026946, 1.82884821782605590326422474657, 3.98269737550828919026532292030, 5.17499475438692113541932398945, 6.60319911388122356928987542434, 7.15362507530846120916278684003, 8.009836374579897754699382611775, 8.746096569896799872026314769927, 10.41536494319635271978262333116, 10.96205998456656464747065496441
