| L(s) = 1 | + (−9 − 9i)3-s + (−15 − 20i)5-s + (−29 + 29i)7-s + 81i·9-s + 118·11-s + (69 + 69i)13-s + (−45 + 315i)15-s + (−271 + 271i)17-s + 280i·19-s + 522·21-s + (−269 − 269i)23-s + (−175 + 600i)25-s − 680i·29-s − 202·31-s + (−1.06e3 − 1.06e3i)33-s + ⋯ |
| L(s) = 1 | + (−1 − i)3-s + (−0.599 − 0.800i)5-s + (−0.591 + 0.591i)7-s + i·9-s + 0.975·11-s + (0.408 + 0.408i)13-s + (−0.200 + 1.39i)15-s + (−0.937 + 0.937i)17-s + 0.775i·19-s + 1.18·21-s + (−0.508 − 0.508i)23-s + (−0.280 + 0.960i)25-s − 0.808i·29-s − 0.210·31-s + (−0.975 − 0.975i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0898 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.151950 + 0.166267i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.151950 + 0.166267i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (15 + 20i)T \) |
| good | 3 | \( 1 + (9 + 9i)T + 81iT^{2} \) |
| 7 | \( 1 + (29 - 29i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 118T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-69 - 69i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (271 - 271i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 - 280iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (269 + 269i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + 680iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 202T + 9.23e5T^{2} \) |
| 37 | \( 1 + (651 - 651i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.68e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (1.08e3 + 1.08e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.26e3 - 1.26e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (611 + 611i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 1.16e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.59e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-751 + 751i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 6.44e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.95e3 + 2.95e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 1.05e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-6.23e3 - 6.23e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 1.44e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (7.31e3 - 7.31e3i)T - 8.85e7iT^{2} \) |
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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
−13.55389769610803420922316558976, −12.49126617138132491543014476262, −12.07378486299941769060545815803, −11.05593650248195123569687280994, −9.290220622696910056336496433318, −8.163420559372984734976821523672, −6.65183338143521544869367835874, −5.87092145649695077088201648484, −4.12267476571919995580683836130, −1.51092581753396996748357753116,
0.13291369947476513892908659453, 3.44493354293145249964170749341, 4.58002665725621892254163500499, 6.21361731335024972686180708857, 7.22721269371798772990754253347, 9.153192522464178186618093036208, 10.29090031740910862929887036134, 11.11668777889133405468566666195, 11.82071930798446062294260502206, 13.40277315145825474034463075286
