| L(s) = 1 | + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (−0.980 + 0.0857i)5-s + (1.16 + 0.981i)7-s + (0.965 + 0.258i)8-s + (0.492 − 0.852i)10-s + (−3.02 − 5.24i)11-s + (−1.58 − 0.737i)13-s + (−1.47 + 0.395i)14-s + (−0.766 + 0.642i)16-s + (−7.17 + 3.34i)17-s + (2.99 − 2.09i)19-s + (0.415 + 0.892i)20-s + (6.03 + 0.527i)22-s + (0.211 + 0.788i)23-s + ⋯ |
| L(s) = 1 | + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (−0.438 + 0.0383i)5-s + (0.442 + 0.370i)7-s + (0.341 + 0.0915i)8-s + (0.155 − 0.269i)10-s + (−0.912 − 1.58i)11-s + (−0.438 − 0.204i)13-s + (−0.394 + 0.105i)14-s + (−0.191 + 0.160i)16-s + (−1.73 + 0.811i)17-s + (0.686 − 0.480i)19-s + (0.0930 + 0.199i)20-s + (1.28 + 0.112i)22-s + (0.0440 + 0.164i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.230413 - 0.309087i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.230413 - 0.309087i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.573 - 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (1.29 + 5.94i)T \) |
| good | 5 | \( 1 + (0.980 - 0.0857i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.16 - 0.981i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (3.02 + 5.24i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.58 + 0.737i)T + (8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (7.17 - 3.34i)T + (10.9 - 13.0i)T^{2} \) |
| 19 | \( 1 + (-2.99 + 2.09i)T + (6.49 - 17.8i)T^{2} \) |
| 23 | \( 1 + (-0.211 - 0.788i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.314 + 1.17i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-0.312 + 0.312i)T - 31iT^{2} \) |
| 41 | \( 1 + (-8.84 + 3.21i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (4.57 + 4.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (11.5 + 6.65i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.93 + 9.45i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.977 - 11.1i)T + (-58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (-3.22 + 6.92i)T + (-39.2 - 46.7i)T^{2} \) |
| 67 | \( 1 + (3.99 - 4.75i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-7.37 - 1.30i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 - 8.51iT - 73T^{2} \) |
| 79 | \( 1 + (0.00716 + 0.0819i)T + (-77.7 + 13.7i)T^{2} \) |
| 83 | \( 1 + (-0.0951 + 0.261i)T + (-63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.92 - 0.867i)T + (87.6 + 15.4i)T^{2} \) |
| 97 | \( 1 + (7.26 - 1.94i)T + (84.0 - 48.5i)T^{2} \) |
| 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
−10.27845031862009435051634769477, −9.128534182294710422387464990149, −8.403487687906678105900024683072, −7.84198068126485993948319319927, −6.77657308065081567083321107843, −5.76024605849049729121589889215, −4.99959616785029701548007685040, −3.67606472589475998613051882824, −2.25133394633432407667715175125, −0.22403953290788382127008550810,
1.78156926111658394727003879454, 2.89178386320671643093184415986, 4.48445539914955297343482881874, 4.79266251167380651557144879146, 6.59125928271868414448353490922, 7.58498351729326716483214917726, 7.990852244322436213660419960683, 9.336991465013859472806769224691, 9.832036270128519026420526998197, 10.83637985597483760046643269550
