| L(s) = 1 | − 2-s − 3-s + 4-s + 2.52·5-s + 6-s + 0.198·7-s − 8-s + 9-s − 2.52·10-s − 1.31·11-s − 12-s + 2.11·13-s − 0.198·14-s − 2.52·15-s + 16-s − 0.387·17-s − 18-s + 19-s + 2.52·20-s − 0.198·21-s + 1.31·22-s − 9.31·23-s + 24-s + 1.39·25-s − 2.11·26-s − 27-s + 0.198·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.13·5-s + 0.408·6-s + 0.0749·7-s − 0.353·8-s + 0.333·9-s − 0.799·10-s − 0.395·11-s − 0.288·12-s + 0.586·13-s − 0.0529·14-s − 0.652·15-s + 0.250·16-s − 0.0940·17-s − 0.235·18-s + 0.229·19-s + 0.565·20-s − 0.0432·21-s + 0.279·22-s − 1.94·23-s + 0.204·24-s + 0.278·25-s − 0.415·26-s − 0.192·27-s + 0.0374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 2.52T + 5T^{2} \) |
| 7 | \( 1 - 0.198T + 7T^{2} \) |
| 11 | \( 1 + 1.31T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 0.387T + 17T^{2} \) |
| 23 | \( 1 + 9.31T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 + 1.48T + 31T^{2} \) |
| 37 | \( 1 + 0.617T + 37T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 + 3.82T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 59 | \( 1 + 3.13T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 - 1.21T + 79T^{2} \) |
| 83 | \( 1 + 4.77T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.019293720895179174140105803547, −6.76618200048420813574130493600, −6.42304865999598752800440955123, −5.70027500945635489953323880250, −5.10171933318439547156850566672, −4.08220464966744058018212385648, −2.99651240724733272094320045991, −2.01327034598780239047156986925, −1.35837345284929925121007389616, 0,
1.35837345284929925121007389616, 2.01327034598780239047156986925, 2.99651240724733272094320045991, 4.08220464966744058018212385648, 5.10171933318439547156850566672, 5.70027500945635489953323880250, 6.42304865999598752800440955123, 6.76618200048420813574130493600, 8.019293720895179174140105803547
