L(s) = 1 | − 1.41·2-s + (1 − 1.41i)3-s − i·5-s + (−1.41 + 2.00i)6-s + i·7-s + 2.82·8-s + (−1.00 − 2.82i)9-s + 1.41i·10-s + (3 − 1.41i)11-s − 2.24i·13-s − 1.41i·14-s + (−1.41 − i)15-s − 4.00·16-s + 5.82·17-s + (1.41 + 4.00i)18-s + 3.24i·19-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (0.577 − 0.816i)3-s − 0.447i·5-s + (−0.577 + 0.816i)6-s + 0.377i·7-s + 0.999·8-s + (−0.333 − 0.942i)9-s + 0.447i·10-s + (0.904 − 0.426i)11-s − 0.621i·13-s − 0.377i·14-s + (−0.365 − 0.258i)15-s − 1.00·16-s + 1.41·17-s + (0.333 + 0.942i)18-s + 0.743i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.117190420\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.117190420\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1 + 1.41i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (-3 + 1.41i)T \) |
good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 17 | \( 1 - 5.82T + 17T^{2} \) |
| 19 | \( 1 - 3.24iT - 19T^{2} \) |
| 23 | \( 1 + 1.24iT - 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 3.24iT - 43T^{2} \) |
| 47 | \( 1 + 1.07iT - 47T^{2} \) |
| 53 | \( 1 + 4.41iT - 53T^{2} \) |
| 59 | \( 1 + 12.8iT - 59T^{2} \) |
| 61 | \( 1 + 9.24iT - 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 1.75iT - 71T^{2} \) |
| 73 | \( 1 - 6.48iT - 73T^{2} \) |
| 79 | \( 1 - 6.24iT - 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 16.4iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.555899756604800097876348000933, −8.589958142248287413505195966165, −8.100204746356944314168026418320, −7.53434469522262184290789923136, −6.36009154275149015037305410084, −5.54190981004119800246563483465, −4.16966036065633226645137977491, −3.08554718656670971787664572559, −1.64577902834960354178380023258, −0.76616304940933317296983778997,
1.32407585073788874643318508836, 2.74287510249239985092247673313, 3.99653908073431237059184813748, 4.53194974454485308280851892249, 5.83748068987371982565615334786, 7.16480524141243564210546644592, 7.65732229056451201190089218398, 8.608008425566825335152677444655, 9.377287095392805596290390304767, 9.763213999047624154582748680121