| L(s) = 1 | − 2·11-s + 13-s − 17-s − 8·19-s + 6·23-s − 4·31-s − 2·37-s − 4·41-s + 4·43-s − 7·49-s − 6·53-s − 4·59-s + 8·61-s − 8·67-s + 4·71-s + 10·79-s + 8·83-s − 6·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s + 0.277·13-s − 0.242·17-s − 1.83·19-s + 1.25·23-s − 0.718·31-s − 0.328·37-s − 0.624·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.520·59-s + 1.02·61-s − 0.977·67-s + 0.474·71-s + 1.12·79-s + 0.878·83-s − 0.635·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6425360423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6425360423\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.61551110204013, −12.05397554952454, −11.35784968666071, −11.06823438418467, −10.69390486888237, −10.30559627048307, −9.760561533689311, −9.139994076909742, −8.866573831754931, −8.367094233575685, −7.946946008463871, −7.397223229742987, −6.923418473513332, −6.368579517366407, −6.135406812322553, −5.387735395484326, −4.891346216311190, −4.630402425562240, −3.795622394159432, −3.558335844906783, −2.742799523538620, −2.366991813238423, −1.727751137869027, −1.133568305087184, −0.2073170442129504,
0.2073170442129504, 1.133568305087184, 1.727751137869027, 2.366991813238423, 2.742799523538620, 3.558335844906783, 3.795622394159432, 4.630402425562240, 4.891346216311190, 5.387735395484326, 6.135406812322553, 6.368579517366407, 6.923418473513332, 7.397223229742987, 7.946946008463871, 8.367094233575685, 8.866573831754931, 9.139994076909742, 9.760561533689311, 10.30559627048307, 10.69390486888237, 11.06823438418467, 11.35784968666071, 12.05397554952454, 12.61551110204013
