L(s) = 1 | + 4.67i·2-s − 3·3-s − 13.8·4-s + 4.61i·5-s − 14.0i·6-s + 11.3i·7-s − 27.3i·8-s + 9·9-s − 21.5·10-s − 37.1·11-s + 41.5·12-s + 1.48·13-s − 52.8·14-s − 13.8i·15-s + 17.1·16-s + 2.35·17-s + ⋯ |
L(s) = 1 | + 1.65i·2-s − 0.577·3-s − 1.73·4-s + 0.413i·5-s − 0.954i·6-s + 0.610i·7-s − 1.20i·8-s + 0.333·9-s − 0.682·10-s − 1.01·11-s + 0.999·12-s + 0.0317·13-s − 1.00·14-s − 0.238i·15-s + 0.267·16-s + 0.0336·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2946163511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2946163511\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 157 | \( 1 + (-1.93e3 + 378. i)T \) |
good | 2 | \( 1 - 4.67iT - 8T^{2} \) |
| 5 | \( 1 - 4.61iT - 125T^{2} \) |
| 7 | \( 1 - 11.3iT - 343T^{2} \) |
| 11 | \( 1 + 37.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 1.48T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.35T + 4.91e3T^{2} \) |
| 19 | \( 1 + 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 110. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 388.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 447. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 96.6iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 532.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 142. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 839. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 841. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 214.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 243. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 688. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 326. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 930.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 801. iT - 9.12e5T^{2} \) |
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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
−10.57762312086954130444590142239, −9.519739866805036143366916920977, −8.440483172115215997651570798037, −7.83258460274182479080576845245, −6.68886858926783320932209487508, −6.17268254817584986793521645643, −5.22231833077601096579590843239, −4.38604008456016998490539212248, −2.52555663539297412729642986964, −0.12057582884268700442170663471,
1.01389444731963975625197542116, 2.25273699191723354448864458327, 3.58535776329493132621475073734, 4.55033773168879653652902642305, 5.44678701830470182555382589026, 6.90181861255975442605249281715, 8.111824789313717258471188103618, 9.176739139953608379022010886294, 10.03402008110639643432412212290, 10.84432537157060179992293753813