L(s) = 1 | + 8.57i·3-s + (−0.582 − 11.1i)5-s + 22.1i·7-s − 46.4·9-s − 27.1·11-s + 70.3i·13-s + (95.7 − 4.99i)15-s − 73.3i·17-s − 110.·19-s − 189.·21-s − 107. i·23-s + (−124. + 13.0i)25-s − 167. i·27-s − 68.6·29-s + 137.·31-s + ⋯ |
L(s) = 1 | + 1.64i·3-s + (−0.0521 − 0.998i)5-s + 1.19i·7-s − 1.72·9-s − 0.743·11-s + 1.50i·13-s + (1.64 − 0.0859i)15-s − 1.04i·17-s − 1.32·19-s − 1.97·21-s − 0.977i·23-s + (−0.994 + 0.104i)25-s − 1.19i·27-s − 0.439·29-s + 0.794·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0233895 - 0.897137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0233895 - 0.897137i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.582 + 11.1i)T \) |
good | 3 | \( 1 - 8.57iT - 27T^{2} \) |
| 7 | \( 1 - 22.1iT - 343T^{2} \) |
| 11 | \( 1 + 27.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 73.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 68.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 137.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 60.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 95.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 501. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 439. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 286. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 547.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 511.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 301. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 82.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 704. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 743.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.13e3iT - 9.12e5T^{2} \) |
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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
−12.76714724840683885827473663433, −11.79682129153654382675471371661, −10.90648166937739243193775782676, −9.601469700127525914461069273188, −9.108716553478851078774929623137, −8.261624042032273344158359608477, −6.13944424062498807066843006031, −4.91649478382524877285923176531, −4.35992245518714077008759898351, −2.54259038736404796202911993060,
0.40005941588809582139455162648, 2.11307239092703246920526119746, 3.55167686017964275347178401325, 5.75671944528856009079135748962, 6.77740362606414402704012015744, 7.60598698782053666539761609174, 8.220861899250727971507641573924, 10.35885014890165992963632355102, 10.77775651946370271005761832899, 12.10924481357144640119429623653