| L(s) = 1 | − 2.66·3-s + 2.00·5-s + 0.318·7-s + 4.11·9-s + 6.07·11-s + 1.85·13-s − 5.36·15-s − 6.02·17-s + 8.28·19-s − 0.848·21-s − 8.04·23-s − 0.960·25-s − 2.97·27-s − 3.94·29-s − 6.44·31-s − 16.2·33-s + 0.639·35-s + 3.56·37-s − 4.94·39-s − 8.28·41-s − 10.3·43-s + 8.27·45-s + 2.95·47-s − 6.89·49-s + 16.0·51-s − 6.64·53-s + 12.2·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s + 0.898·5-s + 0.120·7-s + 1.37·9-s + 1.83·11-s + 0.513·13-s − 1.38·15-s − 1.46·17-s + 1.90·19-s − 0.185·21-s − 1.67·23-s − 0.192·25-s − 0.572·27-s − 0.731·29-s − 1.15·31-s − 2.82·33-s + 0.108·35-s + 0.585·37-s − 0.791·39-s − 1.29·41-s − 1.57·43-s + 1.23·45-s + 0.430·47-s − 0.985·49-s + 2.25·51-s − 0.913·53-s + 1.64·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
| good | 3 | \( 1 + 2.66T + 3T^{2} \) |
| 5 | \( 1 - 2.00T + 5T^{2} \) |
| 7 | \( 1 - 0.318T + 7T^{2} \) |
| 11 | \( 1 - 6.07T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 6.02T + 17T^{2} \) |
| 19 | \( 1 - 8.28T + 19T^{2} \) |
| 23 | \( 1 + 8.04T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 + 6.44T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + 8.28T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 6.64T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 + 4.13T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 2.56T + 79T^{2} \) |
| 83 | \( 1 + 1.05T + 83T^{2} \) |
| 89 | \( 1 + 2.69T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−7.43681006882108432274199227743, −6.71860322627688475928007400910, −6.20013217330551186913090276785, −5.76065891423746732768313527807, −5.01593461941088798777698892194, −4.20218837341249081740957057457, −3.44621054278402075782366944236, −1.83033942542496579150487365423, −1.39131694143639399591009909445, 0,
1.39131694143639399591009909445, 1.83033942542496579150487365423, 3.44621054278402075782366944236, 4.20218837341249081740957057457, 5.01593461941088798777698892194, 5.76065891423746732768313527807, 6.20013217330551186913090276785, 6.71860322627688475928007400910, 7.43681006882108432274199227743
