| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s + 0.618·5-s − 1.61·6-s + 2.23·8-s + 9-s − 1.00·10-s + 2.23·11-s + 0.618·12-s + 2.38·13-s + 0.618·15-s − 4.85·16-s − 6.70·17-s − 1.61·18-s + 3.47·19-s + 0.381·20-s − 3.61·22-s − 23-s + 2.23·24-s − 4.61·25-s − 3.85·26-s + 27-s − 8.23·29-s − 1.00·30-s − 6.70·31-s + 3.38·32-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.276·5-s − 0.660·6-s + 0.790·8-s + 0.333·9-s − 0.316·10-s + 0.674·11-s + 0.178·12-s + 0.660·13-s + 0.159·15-s − 1.21·16-s − 1.62·17-s − 0.381·18-s + 0.796·19-s + 0.0854·20-s − 0.771·22-s − 0.208·23-s + 0.456·24-s − 0.923·25-s − 0.755·26-s + 0.192·27-s − 1.52·29-s − 0.182·30-s − 1.20·31-s + 0.597·32-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 + 6.70T + 17T^{2} \) |
| 19 | \( 1 - 3.47T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 - 7.23T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 9.38T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 9.18T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.465977365885737534832051459393, −7.62610183928608448947655696315, −7.07019578610201141488170700555, −6.22332310970938193118386353079, −5.20876086274239252724403377999, −4.16603769723704442193477873304, −3.51681045943182170705149965664, −2.07422098565908417666595227636, −1.52573085704330742982561786777, 0,
1.52573085704330742982561786777, 2.07422098565908417666595227636, 3.51681045943182170705149965664, 4.16603769723704442193477873304, 5.20876086274239252724403377999, 6.22332310970938193118386353079, 7.07019578610201141488170700555, 7.62610183928608448947655696315, 8.465977365885737534832051459393
