| L(s) = 1 | + i·3-s + i·5-s + 2.67i·7-s − 9-s − 3.76i·11-s + 5.53·13-s − 15-s + (−3.76 − 1.67i)17-s + 6.52·19-s − 2.67·21-s − 7.53i·23-s − 25-s − i·27-s − 4.20i·29-s − 4.75i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 1.00i·7-s − 0.333·9-s − 1.13i·11-s + 1.53·13-s − 0.258·15-s + (−0.914 − 0.405i)17-s + 1.49·19-s − 0.582·21-s − 1.57i·23-s − 0.200·25-s − 0.192i·27-s − 0.781i·29-s − 0.854i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713513987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713513987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 17 | \( 1 + (3.76 + 1.67i)T \) |
| good | 7 | \( 1 - 2.67iT - 7T^{2} \) |
| 11 | \( 1 + 3.76iT - 11T^{2} \) |
| 13 | \( 1 - 5.53T + 13T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 + 7.53iT - 23T^{2} \) |
| 29 | \( 1 + 4.20iT - 29T^{2} \) |
| 31 | \( 1 + 4.75iT - 31T^{2} \) |
| 37 | \( 1 + 4.52iT - 37T^{2} \) |
| 41 | \( 1 + 4.67iT - 41T^{2} \) |
| 43 | \( 1 + 2.75T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 6.06T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 + 7.53iT - 61T^{2} \) |
| 67 | \( 1 - 6.75T + 67T^{2} \) |
| 71 | \( 1 + 11.6iT - 71T^{2} \) |
| 73 | \( 1 + 8.35iT - 73T^{2} \) |
| 79 | \( 1 - 8.75iT - 79T^{2} \) |
| 83 | \( 1 + 4.29T + 83T^{2} \) |
| 89 | \( 1 - 4.19T + 89T^{2} \) |
| 97 | \( 1 - 4.19iT - 97T^{2} \) |
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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
−8.463517156805472667118648277990, −7.910824754157340617333171727873, −6.65997449632528574432719300706, −6.11351968065792712468976975460, −5.53524950408650992969520213603, −4.62303634928054913110375517714, −3.61401768167758330707999920488, −3.03330188011600248018707688185, −2.09642763511171180201313588308, −0.52740966535335234475999600162,
1.24570386335011155845594795395, 1.56519251103622860610843010434, 3.14229687765365992899334932534, 3.83961912695465411229062543778, 4.75111307022711336663610538045, 5.47966107921602214282579768461, 6.48764384135404300157513842658, 7.02214967433171502740155332740, 7.70155805946205716872134615774, 8.365305882473659277727596700853
