L(s) = 1 | − 1.41i·2-s + (−1 − 1.41i)3-s − 2.00·4-s − 4·5-s + (−2.00 + 1.41i)6-s − 2.82i·7-s + 2.82i·8-s + (−1.00 + 2.82i)9-s + 5.65i·10-s − 5.65i·11-s + (2.00 + 2.82i)12-s + 5.65i·13-s − 4.00·14-s + (4 + 5.65i)15-s + 4.00·16-s − 2.82i·17-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (−0.577 − 0.816i)3-s − 1.00·4-s − 1.78·5-s + (−0.816 + 0.577i)6-s − 1.06i·7-s + 1.00i·8-s + (−0.333 + 0.942i)9-s + 1.78i·10-s − 1.70i·11-s + (0.577 + 0.816i)12-s + 1.56i·13-s − 1.06·14-s + (1.03 + 1.46i)15-s + 1.00·16-s − 0.685i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4T + 5T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 2.82iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 2.82iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−10.53685470125070527016905996099, −8.953665078443508240388645990069, −8.230920084641081213512856526707, −7.40168553661686659987483610786, −6.49150672972918407298728380809, −4.89890506441801664898339016546, −4.05671433118967911994455520199, −3.09905732341373504883860162519, −1.14087997679507118761902052691, 0,
3.31234224582318748799997723500, 4.33935575664192211844157380596, 5.00827034516637870928338627933, 6.02896847189947943260690942753, 7.16781318755683762523348533339, 8.017511362948952905068003232191, 8.700517318933019916393915957414, 9.756755912664020849478788683959, 10.58078429484708039132182623579