| L(s) = 1 | − 1.87·2-s − 3.13·3-s + 1.51·4-s − 0.194·5-s + 5.86·6-s + 0.913·8-s + 6.80·9-s + 0.364·10-s + 1.68·11-s − 4.73·12-s + 6.75·13-s + 0.609·15-s − 4.73·16-s + 3.80·17-s − 12.7·18-s − 0.622·19-s − 0.294·20-s − 3.14·22-s − 7.95·23-s − 2.85·24-s − 4.96·25-s − 12.6·26-s − 11.9·27-s + 2.32·29-s − 1.14·30-s + 0.349·31-s + 7.05·32-s + ⋯ |
| L(s) = 1 | − 1.32·2-s − 1.80·3-s + 0.756·4-s − 0.0870·5-s + 2.39·6-s + 0.322·8-s + 2.26·9-s + 0.115·10-s + 0.506·11-s − 1.36·12-s + 1.87·13-s + 0.157·15-s − 1.18·16-s + 0.921·17-s − 3.00·18-s − 0.142·19-s − 0.0658·20-s − 0.671·22-s − 1.65·23-s − 0.583·24-s − 0.992·25-s − 2.48·26-s − 2.29·27-s + 0.431·29-s − 0.208·30-s + 0.0627·31-s + 1.24·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.87T + 2T^{2} \) |
| 3 | \( 1 + 3.13T + 3T^{2} \) |
| 5 | \( 1 + 0.194T + 5T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 6.75T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 + 0.622T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 - 0.349T + 31T^{2} \) |
| 37 | \( 1 + 6.20T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 - 5.24T + 47T^{2} \) |
| 53 | \( 1 + 5.78T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 5.26T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 79 | \( 1 - 7.30T + 79T^{2} \) |
| 83 | \( 1 - 0.0669T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−8.165366806727043405976624463765, −7.55341544011178261650918903783, −6.59626701118678655665810163007, −6.12562651881076674958822495913, −5.47055550271145732712285748384, −4.35742455097783021445762436702, −3.72536990448538072876074552473, −1.71999730335348951553218327339, −1.09610552287818058842386421439, 0,
1.09610552287818058842386421439, 1.71999730335348951553218327339, 3.72536990448538072876074552473, 4.35742455097783021445762436702, 5.47055550271145732712285748384, 6.12562651881076674958822495913, 6.59626701118678655665810163007, 7.55341544011178261650918903783, 8.165366806727043405976624463765
