L(s) = 1 | − 1.47·2-s + i·3-s + 0.182·4-s + (−0.357 − 2.20i)5-s − 1.47i·6-s + 1.62i·7-s + 2.68·8-s − 9-s + (0.528 + 3.26i)10-s − 0.174·11-s + 0.182i·12-s − 4.30·13-s − 2.39i·14-s + (2.20 − 0.357i)15-s − 4.33·16-s + 6.78·17-s + ⋯ |
L(s) = 1 | − 1.04·2-s + 0.577i·3-s + 0.0910·4-s + (−0.159 − 0.987i)5-s − 0.603i·6-s + 0.612i·7-s + 0.949·8-s − 0.333·9-s + (0.167 + 1.03i)10-s − 0.0526·11-s + 0.0525i·12-s − 1.19·13-s − 0.639i·14-s + (0.569 − 0.0923i)15-s − 1.08·16-s + 1.64·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00643 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00643 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.315133 - 0.317169i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.315133 - 0.317169i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.357 + 2.20i)T \) |
| 37 | \( 1 + (-1.01 + 5.99i)T \) |
good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 7 | \( 1 - 1.62iT - 7T^{2} \) |
| 11 | \( 1 + 0.174T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 - 6.78T + 17T^{2} \) |
| 19 | \( 1 + 3.61iT - 19T^{2} \) |
| 23 | \( 1 + 6.61T + 23T^{2} \) |
| 29 | \( 1 + 2.45iT - 29T^{2} \) |
| 31 | \( 1 + 6.18iT - 31T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 11.8iT - 47T^{2} \) |
| 53 | \( 1 + 5.47iT - 53T^{2} \) |
| 59 | \( 1 + 3.00iT - 59T^{2} \) |
| 61 | \( 1 + 11.4iT - 61T^{2} \) |
| 67 | \( 1 + 2.04iT - 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.16iT - 73T^{2} \) |
| 79 | \( 1 - 15.4iT - 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 13.4iT - 89T^{2} \) |
| 97 | \( 1 + 6.23T + 97T^{2} \) |
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
−9.936019190508111248416462236105, −9.800271922243649438504182406508, −8.905354322511808285382467857659, −8.100152067546874233573923544422, −7.45882338111760017763244235619, −5.72806275513381025393800919666, −4.95696620710804462687722889636, −3.95597591454108034409120437826, −2.17890823749720021436167831361, −0.38980693100652251353173943812,
1.39986849854602011368730120269, 2.91717590164728438886521408295, 4.24687758102030480371584097950, 5.70526463038366606489783366641, 6.89472820499923511610880947401, 7.71661644840393079185137842831, 7.956905564744695965000923687106, 9.386682495780910101500117579497, 10.23147023903757749197353743230, 10.51937591269645152362725736068