L(s) = 1 | − 2.67i·2-s − 5.15·4-s + (−1.67 + 1.48i)5-s + 0.806i·7-s + 8.44i·8-s + (3.96 + 4.48i)10-s + 3.67·11-s + i·13-s + 2.15·14-s + 12.2·16-s + 1.35i·17-s + 1.67·19-s + (8.63 − 7.63i)20-s − 9.83i·22-s + 6.48i·23-s + ⋯ |
L(s) = 1 | − 1.89i·2-s − 2.57·4-s + (−0.749 + 0.662i)5-s + 0.304i·7-s + 2.98i·8-s + (1.25 + 1.41i)10-s + 1.10·11-s + 0.277i·13-s + 0.576·14-s + 3.06·16-s + 0.327i·17-s + 0.384·19-s + (1.93 − 1.70i)20-s − 2.09i·22-s + 1.35i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.882300 - 0.397595i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.882300 - 0.397595i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.67 - 1.48i)T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 2.67iT - 2T^{2} \) |
| 7 | \( 1 - 0.806iT - 7T^{2} \) |
| 11 | \( 1 - 3.67T + 11T^{2} \) |
| 17 | \( 1 - 1.35iT - 17T^{2} \) |
| 19 | \( 1 - 1.67T + 19T^{2} \) |
| 23 | \( 1 - 6.48iT - 23T^{2} \) |
| 29 | \( 1 - 2.41T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 - 3.76iT - 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 43 | \( 1 - 6.79iT - 43T^{2} \) |
| 47 | \( 1 + 3.19iT - 47T^{2} \) |
| 53 | \( 1 - 5.73iT - 53T^{2} \) |
| 59 | \( 1 - 5.98T + 59T^{2} \) |
| 61 | \( 1 + 1.76T + 61T^{2} \) |
| 67 | \( 1 - 9.89iT - 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 11.7iT - 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 3.84iT - 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
| 97 | \( 1 + 1.87iT - 97T^{2} \) |
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
−10.85208683009363453562040525887, −9.867563046735222319506349214945, −9.194995284923712443848544276456, −8.321331678040065513823565223499, −7.16172058992983089819421296486, −5.72469226326141289095688224520, −4.33846474712727425608845723228, −3.67585058022720638016301710304, −2.70272030596729739694972576361, −1.33175594654473272691716627954,
0.65263460360565691165992837320, 3.75331338795445326740238017521, 4.48921942771823225494957860959, 5.41931515407399546264626998436, 6.45275597357844479665911850473, 7.26899527139791677796032576689, 7.957182183851543767084872517011, 8.895044026241910514499770671184, 9.306291515719291899092137976240, 10.62295221096288491698068153687