L(s) = 1 | + (3.01 − 1.73i)5-s + (−3.12 − 1.80i)7-s + (1.32 − 2.29i)11-s + (2.36 + 4.09i)13-s − 1.79i·17-s − 4.55i·19-s + (−0.377 − 0.653i)23-s + (3.54 − 6.13i)25-s + (−7.19 − 4.15i)29-s + (1.94 − 1.12i)31-s − 12.5·35-s − 3.98·37-s + (5.57 − 3.22i)41-s + (7.60 + 4.39i)43-s + (−1.37 + 2.38i)47-s + ⋯ |
L(s) = 1 | + (1.34 − 0.777i)5-s + (−1.18 − 0.681i)7-s + (0.399 − 0.691i)11-s + (0.655 + 1.13i)13-s − 0.434i·17-s − 1.04i·19-s + (−0.0787 − 0.136i)23-s + (0.708 − 1.22i)25-s + (−1.33 − 0.771i)29-s + (0.348 − 0.201i)31-s − 2.11·35-s − 0.655·37-s + (0.871 − 0.502i)41-s + (1.15 + 0.669i)43-s + (−0.201 + 0.348i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.31938 - 1.06165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.31938 - 1.06165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-3.01 + 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3.12 + 1.80i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.36 - 4.09i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 + 4.55iT - 19T^{2} \) |
| 23 | \( 1 + (0.377 + 0.653i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (7.19 + 4.15i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.94 + 1.12i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.98T + 37T^{2} \) |
| 41 | \( 1 + (-5.57 + 3.22i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-7.60 - 4.39i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.37 - 2.38i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + (1.36 + 2.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.19 + 2.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.78 - 5.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.0730T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + (-4.51 - 2.60i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.244 - 0.424i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 4.41iT - 89T^{2} \) |
| 97 | \( 1 + (-7.21 + 12.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.556948462959980266253828302596, −9.441616861616118070630254222901, −8.621115480275835560346281916006, −7.21025865046260823544178117992, −6.35914263052040071912611390095, −5.82437296438919580020813457264, −4.60100730362483384766743739851, −3.58377013042574429107476053243, −2.22263297022784838647546993526, −0.838897518136814746470972743790,
1.74332154917715879185751655821, 2.84383201023571177248486209233, 3.71839931525773219753202058260, 5.48978903508518166263088928430, 5.98475954063916217368475334696, 6.65770172676126403050348374665, 7.70972246326939462567357318852, 8.981363026810218401361891989001, 9.549435684579085421803245464983, 10.31088696500775294130925629290