| L(s) = 1 | − 1.87·2-s + 3-s + 1.53·4-s − 2.65·5-s − 1.87·6-s + 0.875·8-s + 9-s + 4.99·10-s + 6.05·11-s + 1.53·12-s + 2.90·13-s − 2.65·15-s − 4.71·16-s + 5.61·17-s − 1.87·18-s + 7.23·19-s − 4.07·20-s − 11.3·22-s − 23-s + 0.875·24-s + 2.05·25-s − 5.45·26-s + 27-s + 8.39·29-s + 4.99·30-s − 6.09·31-s + 7.11·32-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.577·3-s + 0.767·4-s − 1.18·5-s − 0.767·6-s + 0.309·8-s + 0.333·9-s + 1.57·10-s + 1.82·11-s + 0.442·12-s + 0.805·13-s − 0.685·15-s − 1.17·16-s + 1.36·17-s − 0.443·18-s + 1.66·19-s − 0.910·20-s − 2.42·22-s − 0.208·23-s + 0.178·24-s + 0.410·25-s − 1.07·26-s + 0.192·27-s + 1.55·29-s + 0.911·30-s − 1.09·31-s + 1.25·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214556159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214556159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 11 | \( 1 - 6.05T + 11T^{2} \) |
| 13 | \( 1 - 2.90T + 13T^{2} \) |
| 17 | \( 1 - 5.61T + 17T^{2} \) |
| 19 | \( 1 - 7.23T + 19T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 6.09T + 31T^{2} \) |
| 37 | \( 1 - 0.781T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 + 7.77T + 67T^{2} \) |
| 71 | \( 1 - 9.12T + 71T^{2} \) |
| 73 | \( 1 - 0.883T + 73T^{2} \) |
| 79 | \( 1 + 5.91T + 79T^{2} \) |
| 83 | \( 1 + 8.51T + 83T^{2} \) |
| 89 | \( 1 - 8.53T + 89T^{2} \) |
| 97 | \( 1 + 0.669T + 97T^{2} \) |
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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
−8.593613944751105842311415570659, −8.004840011407740780555900063560, −7.42591778294202756692318139743, −6.85884618611389024101394581838, −5.79106449832754374468490792235, −4.47574349022313046146507081906, −3.78059508055630307656838423275, −3.11025418437386516967000604337, −1.46850567810441648637292449420, −0.907483187083727688176828941136,
0.907483187083727688176828941136, 1.46850567810441648637292449420, 3.11025418437386516967000604337, 3.78059508055630307656838423275, 4.47574349022313046146507081906, 5.79106449832754374468490792235, 6.85884618611389024101394581838, 7.42591778294202756692318139743, 8.004840011407740780555900063560, 8.593613944751105842311415570659
