L(s) = 1 | + (−2.99 + 0.143i)3-s − 2.23i·5-s + 2.24·7-s + (8.95 − 0.862i)9-s + 9.03i·11-s − 8.03·13-s + (0.321 + 6.70i)15-s − 7.34i·17-s − 14.0·19-s + (−6.72 + 0.322i)21-s + 12.6i·23-s − 5.00·25-s + (−26.7 + 3.87i)27-s − 38.9i·29-s + 44.0·31-s + ⋯ |
L(s) = 1 | + (−0.998 + 0.0479i)3-s − 0.447i·5-s + 0.320·7-s + (0.995 − 0.0957i)9-s + 0.821i·11-s − 0.617·13-s + (0.0214 + 0.446i)15-s − 0.432i·17-s − 0.737·19-s + (−0.320 + 0.0153i)21-s + 0.548i·23-s − 0.200·25-s + (−0.989 + 0.143i)27-s − 1.34i·29-s + 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0479 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0479 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7915894161\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7915894161\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.99 - 0.143i)T \) |
| 5 | \( 1 + 2.23iT \) |
good | 7 | \( 1 - 2.24T + 49T^{2} \) |
| 11 | \( 1 - 9.03iT - 121T^{2} \) |
| 13 | \( 1 + 8.03T + 169T^{2} \) |
| 17 | \( 1 + 7.34iT - 289T^{2} \) |
| 19 | \( 1 + 14.0T + 361T^{2} \) |
| 23 | \( 1 - 12.6iT - 529T^{2} \) |
| 29 | \( 1 + 38.9iT - 841T^{2} \) |
| 31 | \( 1 - 44.0T + 961T^{2} \) |
| 37 | \( 1 - 25.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.51iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 45.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 51.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 86.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 67.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 15.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 90.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 31.3T + 6.24e3T^{2} \) |
| 83 | \( 1 - 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 140. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 0.825T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.938861006414827189765405889879, −9.549810336117316978364381640083, −8.261918624495212864134662315015, −7.45310428286836245918363427437, −6.58030451880608395497130667912, −5.67003713095908478431339250574, −4.72256846112789231364578575847, −4.22656686366843913230659257493, −2.42041873708816475857734649812, −1.11518655426780327867201120171,
0.32796461653353416137994583632, 1.81372292378995502152570146568, 3.23545790502699252496335295484, 4.47001139793416157517454198614, 5.25692056815266849866139363586, 6.30360155352867484340437193591, 6.79722249492364720202612477843, 7.907102025260862268231036932112, 8.690844267009129557566600486842, 9.968967798287602478113083216024