| L(s) = 1 | − 1.86·2-s + 3-s + 1.48·4-s + 4.30·5-s − 1.86·6-s + 7-s + 0.956·8-s + 9-s − 8.04·10-s + 3.88·11-s + 1.48·12-s − 1.86·14-s + 4.30·15-s − 4.76·16-s + 6.82·17-s − 1.86·18-s − 5.88·19-s + 6.41·20-s + 21-s − 7.26·22-s − 1.28·23-s + 0.956·24-s + 13.5·25-s + 27-s + 1.48·28-s + 1.83·29-s − 8.04·30-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 0.577·3-s + 0.743·4-s + 1.92·5-s − 0.762·6-s + 0.377·7-s + 0.338·8-s + 0.333·9-s − 2.54·10-s + 1.17·11-s + 0.429·12-s − 0.499·14-s + 1.11·15-s − 1.19·16-s + 1.65·17-s − 0.440·18-s − 1.35·19-s + 1.43·20-s + 0.218·21-s − 1.54·22-s − 0.267·23-s + 0.195·24-s + 2.71·25-s + 0.192·27-s + 0.281·28-s + 0.339·29-s − 1.46·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.072118293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.072118293\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 1.28T + 23T^{2} \) |
| 29 | \( 1 - 1.83T + 29T^{2} \) |
| 31 | \( 1 + 1.37T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 6.71T + 41T^{2} \) |
| 43 | \( 1 - 9.00T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 + 3.34T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 7.48T + 67T^{2} \) |
| 71 | \( 1 + 4.78T + 71T^{2} \) |
| 73 | \( 1 - 7.13T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 3.87T + 83T^{2} \) |
| 89 | \( 1 - 9.71T + 89T^{2} \) |
| 97 | \( 1 + 1.63T + 97T^{2} \) |
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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
−8.852044971312876058094270832217, −8.114673776333089437697803943069, −7.19361042626197270603766196710, −6.52612269527229026252814263368, −5.76303225704020556184476311231, −4.86983859500250526026536765267, −3.76966584567200648288192235576, −2.48839800406592479775551719464, −1.72538526068262674054550192839, −1.15181368897146101394923570349,
1.15181368897146101394923570349, 1.72538526068262674054550192839, 2.48839800406592479775551719464, 3.76966584567200648288192235576, 4.86983859500250526026536765267, 5.76303225704020556184476311231, 6.52612269527229026252814263368, 7.19361042626197270603766196710, 8.114673776333089437697803943069, 8.852044971312876058094270832217
