L(s) = 1 | + 0.697·2-s − 3-s − 1.51·4-s − 3.10·5-s − 0.697·6-s − 2.45·8-s + 9-s − 2.16·10-s + 1.68·11-s + 1.51·12-s − 1.74·13-s + 3.10·15-s + 1.31·16-s + 0.100·17-s + 0.697·18-s − 4.73·19-s + 4.69·20-s + 1.17·22-s + 6.23·23-s + 2.45·24-s + 4.61·25-s − 1.21·26-s − 27-s − 6.80·29-s + 2.16·30-s + 0.735·31-s + 5.82·32-s + ⋯ |
L(s) = 1 | + 0.493·2-s − 0.577·3-s − 0.756·4-s − 1.38·5-s − 0.284·6-s − 0.866·8-s + 0.333·9-s − 0.684·10-s + 0.507·11-s + 0.436·12-s − 0.484·13-s + 0.800·15-s + 0.329·16-s + 0.0244·17-s + 0.164·18-s − 1.08·19-s + 1.04·20-s + 0.250·22-s + 1.29·23-s + 0.500·24-s + 0.922·25-s − 0.238·26-s − 0.192·27-s − 1.26·29-s + 0.394·30-s + 0.132·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 - 0.697T + 2T^{2} \) |
| 5 | \( 1 + 3.10T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.100T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 - 6.23T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 - 0.735T + 31T^{2} \) |
| 37 | \( 1 - 5.71T + 37T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 - 9.74T + 47T^{2} \) |
| 53 | \( 1 - 3.28T + 53T^{2} \) |
| 59 | \( 1 - 0.705T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 0.0702T + 73T^{2} \) |
| 79 | \( 1 - 9.33T + 79T^{2} \) |
| 83 | \( 1 + 7.84T + 83T^{2} \) |
| 89 | \( 1 + 2.28T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.60225729730594897710799552683, −7.07460844910018380288770942043, −6.19171705076719063315646082030, −5.44145227014154075459982829928, −4.66933086242759725527899102567, −4.11672226685035845174033175568, −3.62700659588397528587789714020, −2.54974235847051188771886601034, −0.943769022431447146803064647171, 0,
0.943769022431447146803064647171, 2.54974235847051188771886601034, 3.62700659588397528587789714020, 4.11672226685035845174033175568, 4.66933086242759725527899102567, 5.44145227014154075459982829928, 6.19171705076719063315646082030, 7.07460844910018380288770942043, 7.60225729730594897710799552683