L(s) = 1 | + 1.85·2-s + 3-s + 1.44·4-s − 3.73·5-s + 1.85·6-s − 1.02·8-s + 9-s − 6.93·10-s + 5.64·11-s + 1.44·12-s + 1.32·13-s − 3.73·15-s − 4.79·16-s − 4.03·17-s + 1.85·18-s − 7.55·19-s − 5.41·20-s + 10.4·22-s − 23-s − 1.02·24-s + 8.93·25-s + 2.46·26-s + 27-s − 9.72·29-s − 6.93·30-s + 7.95·31-s − 6.86·32-s + ⋯ |
L(s) = 1 | + 1.31·2-s + 0.577·3-s + 0.724·4-s − 1.66·5-s + 0.758·6-s − 0.361·8-s + 0.333·9-s − 2.19·10-s + 1.70·11-s + 0.418·12-s + 0.367·13-s − 0.963·15-s − 1.19·16-s − 0.979·17-s + 0.437·18-s − 1.73·19-s − 1.20·20-s + 2.23·22-s − 0.208·23-s − 0.208·24-s + 1.78·25-s + 0.482·26-s + 0.192·27-s − 1.80·29-s − 1.26·30-s + 1.42·31-s − 1.21·32-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 | \( 1 - 1.85T + 2T^{2} \) |
| 5 | \( 1 + 3.73T + 5T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 4.03T + 17T^{2} \) |
| 19 | \( 1 + 7.55T + 19T^{2} \) |
| 29 | \( 1 + 9.72T + 29T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 - 0.802T + 37T^{2} \) |
| 41 | \( 1 + 6.20T + 41T^{2} \) |
| 43 | \( 1 + 0.213T + 43T^{2} \) |
| 47 | \( 1 + 0.832T + 47T^{2} \) |
| 53 | \( 1 + 5.97T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 9.99T + 71T^{2} \) |
| 73 | \( 1 - 3.36T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 - 1.45T + 89T^{2} \) |
| 97 | \( 1 + 7.93T + 97T^{2} \) |
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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
−8.325605014770471817549600893878, −7.34486276456450872703902512714, −6.58426872211637023266883049793, −6.12424479540986987015541900200, −4.60387599038693879389197001477, −4.30360964867235372109403523331, −3.74587017071873507121096677577, −3.05905584124788354508410806851, −1.78480316280788411719374453686, 0,
1.78480316280788411719374453686, 3.05905584124788354508410806851, 3.74587017071873507121096677577, 4.30360964867235372109403523331, 4.60387599038693879389197001477, 6.12424479540986987015541900200, 6.58426872211637023266883049793, 7.34486276456450872703902512714, 8.325605014770471817549600893878