L(s) = 1 | − i·2-s − i·3-s − 4-s + (−1 − 2i)5-s − 6-s − i·7-s + i·8-s − 9-s + (−2 + i)10-s − 2·11-s + i·12-s + 2i·13-s − 14-s + (−2 + i)15-s + 16-s − 8i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s + (−0.632 + 0.316i)10-s − 0.603·11-s + 0.288i·12-s + 0.554i·13-s − 0.267·14-s + (−0.516 + 0.258i)15-s + 0.250·16-s − 1.94i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.214715 - 0.909549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.214715 - 0.909549i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−11.87516824747386644714238969213, −11.36285221408295252654928188196, −9.976663410368504927093723199896, −9.049255686670733671635867707160, −8.015255240138304758518364371083, −7.05683691550879363480497476428, −5.34915356396605090891664883229, −4.34401055138544843684914808244, −2.70745549179855071060634531176, −0.841117477816508379795875374473,
2.96870758936338633814540348254, 4.24627869961818121660718616196, 5.62030931674211299426613422821, 6.56577000271706956530288027022, 7.896812855572944601763094997567, 8.534746866712033934298779236000, 10.05410291922253338571499584509, 10.55799777997487066299444878398, 11.77637777828850373241937052240, 12.85341074076681028669562334064