L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (1.5 − 2.59i)5-s + (−0.5 − 2.59i)7-s + 0.999·8-s + (1.5 + 2.59i)10-s + (−1.5 − 2.59i)11-s − 13-s + (2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (3.5 − 6.06i)19-s − 3·20-s + 3·22-s + (−4.5 + 7.79i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.670 − 1.16i)5-s + (−0.188 − 0.981i)7-s + 0.353·8-s + (0.474 + 0.821i)10-s + (−0.452 − 0.783i)11-s − 0.277·13-s + (0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.802 − 1.39i)19-s − 0.670·20-s + 0.639·22-s + (−0.938 + 1.62i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041211321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041211321\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9T + 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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
−9.455967414600466782049884595482, −8.858715184913233297577231728170, −7.80248977176881529367216742748, −7.33918464000050529623638159710, −6.03555339202875936953791011893, −5.48282874177802778030289571383, −4.58475787561205409920303095381, −3.44579368893884223304965739463, −1.68832937132653323321807153076, −0.50027483335206789807484504301,
1.90125293647238192678882950015, 2.61916154080112620792118898406, 3.50744770951295960094005131160, 4.96656064777934098147169083377, 5.85774895963751147474830638234, 6.77387675113480571165664383009, 7.62872203459024012761075058413, 8.541776383106467103341770360383, 9.556983611093547240577746528942, 10.08193305818444774546168739273