L(s) = 1 | + (1 − 1.73i)4-s + (−0.5 − 0.866i)5-s + (2.5 + 0.866i)7-s − 13-s + (−1.99 − 3.46i)16-s + (3 − 5.19i)17-s + (−2.5 − 4.33i)19-s − 1.99·20-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + (4 − 3.46i)28-s + 6·29-s + (−2.5 + 4.33i)31-s + (−0.500 − 2.59i)35-s + (3.5 + 6.06i)37-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (−0.223 − 0.387i)5-s + (0.944 + 0.327i)7-s − 0.277·13-s + (−0.499 − 0.866i)16-s + (0.727 − 1.26i)17-s + (−0.573 − 0.993i)19-s − 0.447·20-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + (0.755 − 0.654i)28-s + 1.11·29-s + (−0.449 + 0.777i)31-s + (−0.0845 − 0.439i)35-s + (0.575 + 0.996i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.34361 - 0.666068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.34361 - 0.666068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.5 + 6.06i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−11.56435231601265826339740340824, −10.71032054739340183406463419753, −9.664568004548508713651560743111, −8.781759805368727545139118771297, −7.62568741502757608385075518049, −6.69386668812192113543142770088, −5.29361037150917152580677605651, −4.80731631987292312102069204479, −2.79339907232768841076749241642, −1.26808572845763836649430746814,
1.99930704427235173080527762837, 3.47559153022898873773018368587, 4.49085993190327801605516736560, 6.05381080214921419330021229560, 7.13266009033408784716826599938, 8.016420556772781537181595938986, 8.584220237126951316049148906100, 10.29697573743675917707079220577, 10.83262368802642757133536186580, 11.88914951292933934468641466781