| L(s) = 1 | + i·2-s − 3-s − 4-s − i·5-s − i·6-s + i·7-s − i·8-s + 9-s + 10-s − i·11-s + 12-s + (3 − 2i)13-s − 14-s + i·15-s + 16-s + 17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s − 0.447i·5-s − 0.408i·6-s + 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s − 0.301i·11-s + 0.288·12-s + (0.832 − 0.554i)13-s − 0.267·14-s + 0.258i·15-s + 0.250·16-s + 0.242·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13638 + 0.344069i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13638 + 0.344069i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + iT - 5T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 11iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.78104301170401505307511686094, −10.03956400679909049827172074353, −8.783531309560911449799472382690, −8.383785026984063472299012064678, −7.13626991888380856938724060512, −6.23283583615769451213451930779, −5.42033514195581625818776081159, −4.58595412521107281643708288977, −3.18728483142858966879703378545, −1.03407213013071041944117832355,
1.11881376874371552172747247000, 2.72104849752177574724365641659, 3.98271519982292327097434245597, 4.89211644729518278801671489593, 6.16423625528239770109262574331, 6.98106462795789150955807148679, 8.109639533376156155836635696537, 9.219480901588857273058422341604, 10.02004710064398676615273167938, 11.04505362659265251242170856084
