L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.133 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−1.86 − 1.23i)5-s + (0.366 + 0.366i)6-s + (−0.732 + 2.73i)7-s + 0.999·8-s + (2.36 + 1.36i)9-s + (2 − i)10-s − 2i·11-s + (−0.499 + 0.133i)12-s + (2.23 + 3.86i)13-s + (−1.99 − 2i)14-s + (−0.866 + 0.767i)15-s + (−0.5 + 0.866i)16-s + (3.63 + 2.09i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.0773 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.834 − 0.550i)5-s + (0.149 + 0.149i)6-s + (−0.276 + 1.03i)7-s + 0.353·8-s + (0.788 + 0.455i)9-s + (0.632 − 0.316i)10-s − 0.603i·11-s + (−0.144 + 0.0386i)12-s + (0.619 + 1.07i)13-s + (−0.534 − 0.534i)14-s + (−0.223 + 0.198i)15-s + (−0.125 + 0.216i)16-s + (0.881 + 0.508i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.482 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.881314 + 0.520761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.881314 + 0.520761i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| 37 | \( 1 + (5.5 + 2.59i)T \) |
good | 3 | \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.732 - 2.73i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-2.23 - 3.86i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.63 - 2.09i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.36 - 0.901i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (1.26 + 1.26i)T + 29iT^{2} \) |
| 31 | \( 1 + (1.83 - 1.83i)T - 31iT^{2} \) |
| 41 | \( 1 + (0.401 - 0.232i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + (-4.19 - 4.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (-3.13 - 11.6i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (2.09 + 7.83i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (6.46 + 1.73i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 3.63i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (5.46 + 9.46i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + (6.83 + 1.83i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (3.43 + 12.8i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-12.5 + 3.36i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 5.46iT - 97T^{2} \) |
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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
−11.65251061247566724089060680989, −10.61659865487924380626231554860, −9.319435132109838122293877029716, −8.739780769352797807812822476756, −7.84669274081683740491010001061, −6.97669237460714715250315086268, −5.82956848961958141155756559045, −4.82032778253639872022704033537, −3.46771891406794368239209058427, −1.44099208712006002643621054645,
0.935111060567454286714856075917, 3.21479131348686224121636395301, 3.76554601353263863144179051282, 5.03973352116248190176634666391, 6.97707075817635144948493727973, 7.36364614950166980323497024033, 8.506669435061176224482599103258, 9.797582429897144429768776972096, 10.26212643897124940412880352351, 11.08296130940835369449460601239