| L(s) = 1 | − i·3-s + 2.23·5-s − i·7-s − 9-s − 2·11-s − 4.47i·13-s − 2.23i·15-s − 6.47i·17-s − 2·19-s − 21-s + 4i·23-s + 5.00·25-s + i·27-s + 8.47·29-s − 0.472·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.999·5-s − 0.377i·7-s − 0.333·9-s − 0.603·11-s − 1.24i·13-s − 0.577i·15-s − 1.56i·17-s − 0.458·19-s − 0.218·21-s + 0.834i·23-s + 1.00·25-s + 0.192i·27-s + 1.57·29-s − 0.0847·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16284 - 1.16284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16284 - 1.16284i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47iT - 13T^{2} \) |
| 17 | \( 1 + 6.47iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 2.47iT - 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 6.47iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.52T + 61T^{2} \) |
| 67 | \( 1 + 1.52iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 7.52iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 4.94iT - 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 + 3.52iT - 97T^{2} \) |
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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
−10.10390069609309406535592980114, −9.181265799360286878452353010843, −8.222030067689944223866290614703, −7.37690137205070422677603504915, −6.57367228666664486546658771363, −5.55109962022192007261574490470, −4.91921628684325057328323888360, −3.20102759020820697523946123630, −2.31065762657517686106378851302, −0.812442867008367468110116614184,
1.77713501452550967737636706431, 2.82215098697313774227202747509, 4.22913053223247367133274351789, 5.04163831985906238157859868128, 6.14384248911446889150589561705, 6.61595953124195605584696777172, 8.180675204488866195343252656835, 8.783866224017016351125778467968, 9.636546381673703576866129883457, 10.38880761932777607277792014882
