| L(s) = 1 | − i·2-s + 4-s + 2i·7-s − 3i·8-s − 4·11-s + i·13-s + 2·14-s − 16-s − 4i·17-s − 6·19-s + 4i·22-s + 26-s + 2i·28-s + 4·29-s − 10·31-s − 5i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.5·4-s + 0.755i·7-s − 1.06i·8-s − 1.20·11-s + 0.277i·13-s + 0.534·14-s − 0.250·16-s − 0.970i·17-s − 1.37·19-s + 0.852i·22-s + 0.196·26-s + 0.377i·28-s + 0.742·29-s − 1.79·31-s − 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3900094338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3900094338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.432183105823413964062259850804, −7.53355356612732070911146823778, −6.85377203529618854808121645842, −6.00546090574766662132771827290, −5.22306153333215458357659172584, −4.29257128120266313789371610269, −3.16410410949470309381888198153, −2.50150024631790202981254708072, −1.75016680826516673529248393779, −0.10561412153367324341644205896,
1.68186261123444297073973559222, 2.62286568980532115017951900363, 3.69005198998887494817907210079, 4.68378257064995887687037458761, 5.53546138697624766168303037871, 6.20492204608251247810457418887, 7.00009446406438912127659312753, 7.60234051430765614882148630838, 8.260984650709569597645771600855, 8.842696837328822168070456740050
