| L(s) = 1 | + 0.949·2-s + 2.61·3-s − 1.09·4-s − 5-s + 2.48·6-s − 2.94·8-s + 3.82·9-s − 0.949·10-s + 2.27·11-s − 2.86·12-s − 2.72·13-s − 2.61·15-s − 0.599·16-s + 3.28·17-s + 3.63·18-s − 7.67·19-s + 1.09·20-s + 2.15·22-s + 2.22·23-s − 7.68·24-s + 25-s − 2.58·26-s + 2.15·27-s − 1.72·29-s − 2.48·30-s + 4.57·31-s + 5.31·32-s + ⋯ |
| L(s) = 1 | + 0.671·2-s + 1.50·3-s − 0.548·4-s − 0.447·5-s + 1.01·6-s − 1.04·8-s + 1.27·9-s − 0.300·10-s + 0.684·11-s − 0.827·12-s − 0.755·13-s − 0.674·15-s − 0.149·16-s + 0.797·17-s + 0.856·18-s − 1.76·19-s + 0.245·20-s + 0.460·22-s + 0.464·23-s − 1.56·24-s + 0.200·25-s − 0.507·26-s + 0.415·27-s − 0.321·29-s − 0.453·30-s + 0.822·31-s + 0.939·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.949T + 2T^{2} \) |
| 3 | \( 1 - 2.61T + 3T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 + 2.72T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 - 2.22T + 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 4.57T + 31T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 5.83T + 43T^{2} \) |
| 47 | \( 1 + 7.20T + 47T^{2} \) |
| 53 | \( 1 - 4.39T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 5.86T + 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 + 9.68T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 0.0199T + 89T^{2} \) |
| 97 | \( 1 - 1.76T + 97T^{2} \) |
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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
−7.59173805498690605184553767917, −6.70161978089711754650131544730, −6.07025246526347010593373636776, −4.94286732646973957493546358669, −4.48513756078976447228574042961, −3.73988486719872956862389930584, −3.23647611455552837819215299983, −2.52130697942376624460474596568, −1.51058123615715534220401255239, 0,
1.51058123615715534220401255239, 2.52130697942376624460474596568, 3.23647611455552837819215299983, 3.73988486719872956862389930584, 4.48513756078976447228574042961, 4.94286732646973957493546358669, 6.07025246526347010593373636776, 6.70161978089711754650131544730, 7.59173805498690605184553767917
