| L(s) = 1 | + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s + 9-s − 2·10-s − 2·11-s + 12-s + 2·13-s − 2·15-s + 16-s − 2·17-s + 18-s + 19-s − 2·20-s − 2·22-s − 6·23-s + 24-s − 25-s + 2·26-s + 27-s − 6·29-s − 2·30-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.229·19-s − 0.447·20-s − 0.426·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 1.11·29-s − 0.365·30-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.67362421501049789053186556594, −7.27192092680104103055004672734, −6.28213472202174339190222095846, −5.60202602360814003262777188699, −4.68799611931394800638038578911, −3.94230128480093867602899143101, −3.48545467312738195882606406308, −2.52620078794778443646753150383, −1.62994037104660866297684322261, 0,
1.62994037104660866297684322261, 2.52620078794778443646753150383, 3.48545467312738195882606406308, 3.94230128480093867602899143101, 4.68799611931394800638038578911, 5.60202602360814003262777188699, 6.28213472202174339190222095846, 7.27192092680104103055004672734, 7.67362421501049789053186556594
