L(s) = 1 | + (0.375 − 0.650i)2-s + (−1.16 − 2.01i)3-s + (0.717 + 1.24i)4-s + (−2.13 + 3.69i)5-s − 1.75·6-s + 2.58·8-s + (−1.21 + 2.10i)9-s + (1.60 + 2.77i)10-s + (−2.04 − 3.54i)11-s + (1.67 − 2.89i)12-s − 2.18·13-s + 9.94·15-s + (−0.466 + 0.808i)16-s + (−0.295 − 0.510i)17-s + (0.914 + 1.58i)18-s + (0.5 − 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.265 − 0.459i)2-s + (−0.673 − 1.16i)3-s + (0.358 + 0.621i)4-s + (−0.953 + 1.65i)5-s − 0.714·6-s + 0.912·8-s + (−0.405 + 0.703i)9-s + (0.506 + 0.877i)10-s + (−0.617 − 1.07i)11-s + (0.483 − 0.836i)12-s − 0.606·13-s + 2.56·15-s + (−0.116 + 0.202i)16-s + (−0.0715 − 0.123i)17-s + (0.215 + 0.373i)18-s + (0.114 − 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0155210 + 0.244580i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0155210 + 0.244580i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.375 + 0.650i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.16 + 2.01i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.13 - 3.69i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.04 + 3.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.18T + 13T^{2} \) |
| 17 | \( 1 + (0.295 + 0.510i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.18 + 3.78i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 + (2.25 + 3.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.47 - 7.75i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 + 8.69T + 43T^{2} \) |
| 47 | \( 1 + (-5.91 + 10.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.20 + 3.81i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.35 + 9.28i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.21 + 2.10i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 4.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.32T + 71T^{2} \) |
| 73 | \( 1 + (-5.33 - 9.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.28 - 7.41i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 + (1.89 - 3.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−10.08082161009600188397842326360, −8.287754296493122079638870735547, −7.76229788351195735820870416487, −6.86063980417218397899750040385, −6.71301585760037029016010575808, −5.36368490532816578664045401884, −3.82733414451125363010220357371, −3.04466132089058377501540737611, −2.15913849621838037905987404340, −0.10917449168433664247225474344,
1.65399495196012351077920420198, 3.82552377078413233923923787244, 4.67648640361932658879227315379, 5.11844201039040443320275633986, 5.68934452650995156205550137770, 7.27476294747190006719053681421, 7.75278358524055467404411041959, 9.113407319584762276061700187769, 9.601995176760714432172912043798, 10.53087450484822517506033155486