| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s + 4i·7-s + i·8-s − 9-s − 2·11-s − i·12-s + 4·14-s + 16-s + 2i·17-s + i·18-s − 4·21-s + 2i·22-s − i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s + 1.51i·7-s + 0.353i·8-s − 0.333·9-s − 0.603·11-s − 0.288i·12-s + 1.06·14-s + 0.250·16-s + 0.485i·17-s + 0.235i·18-s − 0.872·21-s + 0.426i·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6826453941\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6826453941\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.048310987488586818230960361768, −8.369749158830849622726163695797, −7.76860115865797413948480623479, −6.36392212570722653978851548852, −5.80110155840610961592399105441, −4.95330297041993171173896061820, −4.36831178494353495923880787683, −3.09173744627403608460008293702, −2.69721655858785123884089056511, −1.57849636336488535884124184564,
0.21512056250361237315298819939, 1.24595322760943312660261276553, 2.64156845921894809913343354082, 3.72083375007140684079021589082, 4.49330831004622263226039870989, 5.33060200352736723320980304836, 6.19594758454851536533325490384, 6.94609393656072801726459869821, 7.54491682378360765865229958244, 7.889023824608979117311818396722
