L(s) = 1 | + (0.366 − 1.36i)2-s − 1.73i·3-s + (−1.73 − i)4-s + (−2.36 − 0.633i)6-s + (−2 + 1.73i)7-s + (−2 + 1.99i)8-s + 3.73i·11-s + (−1.73 + 2.99i)12-s − 6.46·13-s + (1.63 + 3.36i)14-s + (1.99 + 3.46i)16-s + 0.464·17-s − 6·19-s + (2.99 + 3.46i)21-s + (5.09 + 1.36i)22-s − 5.46·23-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s − 0.999i·3-s + (−0.866 − 0.5i)4-s + (−0.965 − 0.258i)6-s + (−0.755 + 0.654i)7-s + (−0.707 + 0.707i)8-s + 1.12i·11-s + (−0.499 + 0.866i)12-s − 1.79·13-s + (0.436 + 0.899i)14-s + (0.499 + 0.866i)16-s + 0.112·17-s − 1.37·19-s + (0.654 + 0.755i)21-s + (1.08 + 0.291i)22-s − 1.13·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (-0.366 + 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 3 | \( 1 + 1.73iT - 3T^{2} \) |
| 11 | \( 1 - 3.73iT - 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 0.464T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 2.53iT - 37T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 1.73iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 2.53iT - 61T^{2} \) |
| 67 | \( 1 + 3.46T + 67T^{2} \) |
| 71 | \( 1 - 0.535iT - 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.66iT - 79T^{2} \) |
| 83 | \( 1 + 8.53iT - 83T^{2} \) |
| 89 | \( 1 - 9.46iT - 89T^{2} \) |
| 97 | \( 1 - 7.39T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.887785423334069428680837216268, −9.177021665471622832984837584184, −8.032776293192002675944300816606, −7.06959897838279528725988942295, −6.20995323802689256026159277884, −5.05001172950635862879963783820, −4.06706948602700931474719011142, −2.48450189962822176728262392490, −1.99889863511187345292225532866, 0,
2.99605312371938134764683502880, 4.04404603261967170687404738390, 4.67885885569612435376733762452, 5.78720637507001882686884694114, 6.68354836189646386812100605066, 7.56120126439705187181673483387, 8.529981017220174318730891235364, 9.486610938172564060555778395670, 10.04360490320481575047825233965