L(s) = 1 | + (−0.105 + 0.0611i)2-s + (−0.992 + 1.71i)4-s + (−0.264 + 0.458i)5-s − 0.487i·8-s − 0.0647i·10-s + (3.64 − 2.10i)11-s + (1.74 + 1.00i)13-s + (−1.95 − 3.38i)16-s + 4.38·17-s + 5.24i·19-s + (−0.525 − 0.910i)20-s + (−0.257 + 0.445i)22-s + (−5.43 − 3.13i)23-s + (2.35 + 4.08i)25-s − 0.246·26-s + ⋯ |
L(s) = 1 | + (−0.0749 + 0.0432i)2-s + (−0.496 + 0.859i)4-s + (−0.118 + 0.205i)5-s − 0.172i·8-s − 0.0204i·10-s + (1.09 − 0.633i)11-s + (0.484 + 0.279i)13-s + (−0.488 − 0.846i)16-s + 1.06·17-s + 1.20i·19-s + (−0.117 − 0.203i)20-s + (−0.0548 + 0.0949i)22-s + (−1.13 − 0.654i)23-s + (0.471 + 0.817i)25-s − 0.0484·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.441925391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.441925391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.105 - 0.0611i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (0.264 - 0.458i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.64 + 2.10i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.74 - 1.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 - 5.24iT - 19T^{2} \) |
| 23 | \( 1 + (5.43 + 3.13i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.27 + 4.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.03 + 0.595i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + (0.0994 - 0.172i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 6.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.98 - 8.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.21iT - 53T^{2} \) |
| 59 | \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (11.3 - 6.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.29 + 5.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.50iT - 71T^{2} \) |
| 73 | \( 1 - 5.61iT - 73T^{2} \) |
| 79 | \( 1 + (0.286 + 0.495i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.42 - 9.39i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + (0.493 - 0.285i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.655908115359556209794154485266, −8.903084271218048268985632471618, −8.171281569511352258408167635735, −7.57123210001549462839533535827, −6.47006636748589106827961407888, −5.79062154230297953875273552059, −4.39697854464597927179241686722, −3.76952725716523415317636945946, −2.90978575823178572331755255584, −1.20543459476724308925110979509,
0.76265963553010480818212108176, 1.86793819239261347824714592913, 3.43929726142377184511071722764, 4.44594093938565305098977334675, 5.18467510994006877493549515273, 6.14872240266310933596003248655, 6.87331458522389257116768464964, 7.969239115028263744524011046078, 8.899502881747630379754279546525, 9.358557968766731845763276505169