| L(s) = 1 | + (−0.851 + 0.523i)2-s + (−1.04 + 2.79i)3-s + (0.451 − 0.892i)4-s + (−0.00420 − 0.00572i)5-s + (−0.571 − 2.92i)6-s + (0.656 − 1.00i)7-s + (0.0825 + 0.996i)8-s + (−4.44 − 3.86i)9-s + (0.00657 + 0.00267i)10-s + (1.98 + 0.681i)11-s + (2.01 + 2.19i)12-s + (−2.28 − 2.77i)13-s + (−0.0330 + 1.19i)14-s + (0.0203 − 0.00576i)15-s + (−0.592 − 0.805i)16-s + (−1.58 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.602 + 0.370i)2-s + (−0.603 + 1.61i)3-s + (0.225 − 0.446i)4-s + (−0.00188 − 0.00255i)5-s + (−0.233 − 1.19i)6-s + (0.248 − 0.379i)7-s + (0.0291 + 0.352i)8-s + (−1.48 − 1.28i)9-s + (0.00208 + 0.000845i)10-s + (0.598 + 0.205i)11-s + (0.582 + 0.633i)12-s + (−0.634 − 0.770i)13-s + (−0.00883 + 0.320i)14-s + (0.00526 − 0.00148i)15-s + (−0.148 − 0.201i)16-s + (−0.383 − 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.527541 - 0.109892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.527541 - 0.109892i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.851 - 0.523i)T \) |
| 19 | \( 1 + (0.267 + 4.35i)T \) |
| good | 3 | \( 1 + (1.04 - 2.79i)T + (-2.26 - 1.97i)T^{2} \) |
| 5 | \( 1 + (0.00420 + 0.00572i)T + (-1.49 + 4.77i)T^{2} \) |
| 7 | \( 1 + (-0.656 + 1.00i)T + (-2.81 - 6.41i)T^{2} \) |
| 11 | \( 1 + (-1.98 - 0.681i)T + (8.68 + 6.75i)T^{2} \) |
| 13 | \( 1 + (2.28 + 2.77i)T + (-2.49 + 12.7i)T^{2} \) |
| 17 | \( 1 + (1.58 + 3.12i)T + (-10.0 + 13.6i)T^{2} \) |
| 23 | \( 1 + (2.14 + 5.72i)T + (-17.3 + 15.1i)T^{2} \) |
| 29 | \( 1 + (5.75 + 0.317i)T + (28.8 + 3.19i)T^{2} \) |
| 31 | \( 1 + (-1.43 + 0.777i)T + (16.9 - 25.9i)T^{2} \) |
| 37 | \( 1 + (6.92 + 2.37i)T + (29.1 + 22.7i)T^{2} \) |
| 41 | \( 1 + (-6.75 - 6.57i)T + (1.12 + 40.9i)T^{2} \) |
| 43 | \( 1 + (-3.70 + 1.50i)T + (30.8 - 29.9i)T^{2} \) |
| 47 | \( 1 + (0.494 + 2.53i)T + (-43.5 + 17.6i)T^{2} \) |
| 53 | \( 1 + (-2.19 - 11.2i)T + (-49.1 + 19.9i)T^{2} \) |
| 59 | \( 1 + (7.81 + 7.60i)T + (1.62 + 58.9i)T^{2} \) |
| 61 | \( 1 + (2.93 - 1.38i)T + (38.7 - 47.0i)T^{2} \) |
| 67 | \( 1 + (4.23 - 2.93i)T + (23.4 - 62.7i)T^{2} \) |
| 71 | \( 1 + (-10.0 - 4.73i)T + (45.1 + 54.8i)T^{2} \) |
| 73 | \( 1 + (6.10 + 12.0i)T + (-43.2 + 58.8i)T^{2} \) |
| 79 | \( 1 + (-0.782 + 0.317i)T + (56.6 - 55.0i)T^{2} \) |
| 83 | \( 1 + (-3.72 - 8.48i)T + (-56.2 + 61.0i)T^{2} \) |
| 89 | \( 1 + (-4.70 + 9.30i)T + (-52.7 - 71.7i)T^{2} \) |
| 97 | \( 1 + (7.06 + 4.89i)T + (34.0 + 90.8i)T^{2} \) |
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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
−10.44034726157985752243111542254, −9.427826393503453779517122684632, −9.035502237531901613739743345726, −7.84338452262936514995489959844, −6.78394977604457438684297726768, −5.80821442883682754853469635529, −4.80222001987949815381495332260, −4.24175194123151921886262098974, −2.73420377740734098099359446627, −0.37441098547301315984014729960,
1.45646339871563715186978607992, 2.09688830768956308009916945709, 3.73600301943852843816832548197, 5.40278963747902782471387742215, 6.23336147911252831778634704802, 7.12390236090581733496913948801, 7.73756847308195060685859724220, 8.665678545555473253633898866087, 9.480930556277201057042084378415, 10.71696559071171379312326636119
