L(s) = 1 | + (−4.89 + 1.73i)3-s + 16.9i·5-s + 17.3i·7-s + (20.9 − 16.9i)9-s − 29.3·11-s + 26·13-s + (−29.3 − 83.1i)15-s + 67.8i·17-s − 107. i·19-s + (−30 − 84.8i)21-s − 176.·23-s − 162.·25-s + (−73.4 + 119. i)27-s − 16.9i·29-s + 31.1i·31-s + ⋯ |
L(s) = 1 | + (−0.942 + 0.333i)3-s + 1.51i·5-s + 0.935i·7-s + (0.777 − 0.628i)9-s − 0.805·11-s + 0.554·13-s + (−0.505 − 1.43i)15-s + 0.968i·17-s − 1.29i·19-s + (−0.311 − 0.881i)21-s − 1.59·23-s − 1.30·25-s + (−0.523 + 0.851i)27-s − 0.108i·29-s + 0.180i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0942736 - 0.549466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0942736 - 0.549466i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.89 - 1.73i)T \) |
good | 5 | \( 1 - 16.9iT - 125T^{2} \) |
| 7 | \( 1 - 17.3iT - 343T^{2} \) |
| 11 | \( 1 + 29.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.8iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 107. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 176.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 31.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 206T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 93.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 50.9iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 558.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 278T + 2.26e5T^{2} \) |
| 67 | \( 1 - 890. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 58.7T + 3.57e5T^{2} \) |
| 73 | \( 1 + 422T + 3.89e5T^{2} \) |
| 79 | \( 1 - 668. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 29.3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 373. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.07e3T + 9.12e5T^{2} \) |
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
−12.37251512002616778523585111187, −11.48203687794604839049218485780, −10.67180690728042434204861446892, −10.08263429154997637120619718297, −8.667010718951912131482805044013, −7.23858673239299916003238330386, −6.26134310222814292622176743740, −5.46080414388818819171037104086, −3.81693371236585325364567291328, −2.37294450124460525331379994650,
0.27973072640983627071548291840, 1.50355090523781583354615996836, 4.10167408959360952729375887294, 5.09109086617813649083046122909, 6.05543776285219732014035139823, 7.51465416148392328939777908135, 8.286059964108047490627256026812, 9.743204941133572199936213886781, 10.55598659633986751871479449524, 11.74599328131394009884232945925