L(s) = 1 | + 2.82i·2-s − 10.8i·3-s − 8.00·4-s − 26.2·5-s + 30.6·6-s − 86.0·7-s − 22.6i·8-s − 36.2·9-s − 74.1i·10-s + 114.·11-s + 86.6i·12-s − 231. i·13-s − 243. i·14-s + 284. i·15-s + 64.0·16-s − 244.·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.20i·3-s − 0.500·4-s − 1.04·5-s + 0.850·6-s − 1.75·7-s − 0.353i·8-s − 0.447·9-s − 0.741i·10-s + 0.948·11-s + 0.601i·12-s − 1.37i·13-s − 1.24i·14-s + 1.26i·15-s + 0.250·16-s − 0.845·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.179290 - 0.436726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.179290 - 0.436726i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2.82iT \) |
| 19 | \( 1 + (-253. - 256. i)T \) |
good | 3 | \( 1 + 10.8iT - 81T^{2} \) |
| 5 | \( 1 + 26.2T + 625T^{2} \) |
| 7 | \( 1 + 86.0T + 2.40e3T^{2} \) |
| 11 | \( 1 - 114.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 231. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 244.T + 8.35e4T^{2} \) |
| 23 | \( 1 + 269.T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.13e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.03e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 302. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 1.76e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.31e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 835.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 656. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.92e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.57e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.00e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 4.10e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 7.01e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.26e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.02e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 3.74e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.30e4iT - 8.85e7T^{2} \) |
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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
−15.39608429293913697667127134715, −13.73923621731755663904761526506, −12.77968769792361660589213327138, −11.97491907783936241633373337875, −9.875544502105352605620330510673, −8.244431150345076049044980754739, −7.11650355526117091921472838055, −6.17309335710984374026919559034, −3.58477140934765143233924250986, −0.31947384406043978224097145339,
3.45850268069678081648832984730, 4.34314650538667650447278291589, 6.75383165481682875017147314444, 9.097453835441313350267248816136, 9.678210252704333080779396324525, 11.13524612182684745054822787881, 12.10200708138997941326495708674, 13.51011563330840567110664098474, 15.04798910823789950301705333879, 16.07992495822755812302467286391