| L(s) = 1 | − 3.11·3-s − 5-s − 4.75·7-s + 6.70·9-s − 2.94·11-s − 0.885·13-s + 3.11·15-s − 1.47·17-s − 2.22·19-s + 14.8·21-s + 3.11·23-s + 25-s − 11.5·27-s + 29-s + 4.18·31-s + 9.17·33-s + 4.75·35-s + 7.51·37-s + 2.75·39-s − 0.945·41-s − 3.70·43-s − 6.70·45-s + 3.17·47-s + 15.6·49-s + 4.58·51-s − 13.9·53-s + 2.94·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.447·5-s − 1.79·7-s + 2.23·9-s − 0.888·11-s − 0.245·13-s + 0.804·15-s − 0.357·17-s − 0.511·19-s + 3.23·21-s + 0.649·23-s + 0.200·25-s − 2.21·27-s + 0.185·29-s + 0.752·31-s + 1.59·33-s + 0.804·35-s + 1.23·37-s + 0.441·39-s − 0.147·41-s − 0.564·43-s − 0.999·45-s + 0.463·47-s + 2.23·49-s + 0.642·51-s − 1.91·53-s + 0.397·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 + 4.75T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 0.885T + 13T^{2} \) |
| 17 | \( 1 + 1.47T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 + 0.945T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 - 3.17T + 47T^{2} \) |
| 53 | \( 1 + 13.9T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 4.62T + 61T^{2} \) |
| 67 | \( 1 + 0.824T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 + 6.54T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−7.67397998760267922734187676250, −6.92546531157746883602545208607, −6.44728716374930378264544824447, −5.89575942540403462545445872874, −5.05876627370276910703066977870, −4.40545129523458874167844449342, −3.45409959331407280197943160802, −2.47898758537495672257275580187, −0.789992165931417187187688546857, 0,
0.789992165931417187187688546857, 2.47898758537495672257275580187, 3.45409959331407280197943160802, 4.40545129523458874167844449342, 5.05876627370276910703066977870, 5.89575942540403462545445872874, 6.44728716374930378264544824447, 6.92546531157746883602545208607, 7.67397998760267922734187676250
