| L(s) = 1 | − 3-s − 4·7-s + 9-s + 11-s − 4·13-s + 17-s + 4·21-s + 2·23-s − 27-s + 9·29-s − 8·31-s − 33-s − 2·37-s + 4·39-s − 6·41-s + 12·43-s + 3·47-s + 9·49-s − 51-s − 9·53-s − 3·59-s − 61-s − 4·63-s − 12·67-s − 2·69-s − 6·71-s − 13·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.242·17-s + 0.872·21-s + 0.417·23-s − 0.192·27-s + 1.67·29-s − 1.43·31-s − 0.174·33-s − 0.328·37-s + 0.640·39-s − 0.937·41-s + 1.82·43-s + 0.437·47-s + 9/7·49-s − 0.140·51-s − 1.23·53-s − 0.390·59-s − 0.128·61-s − 0.503·63-s − 1.46·67-s − 0.240·69-s − 0.712·71-s − 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−13.22577844520086, −12.50236286304347, −12.20545606232770, −12.14749830690214, −11.27926939170328, −10.83818323625029, −10.29608090459566, −9.989688069841055, −9.519034360985008, −9.010704485871061, −8.721050633628031, −7.779049793226128, −7.358242564132921, −7.007887415607001, −6.423465439139560, −6.070694704842309, −5.563336655535242, −4.934528993182459, −4.473266123292480, −3.891412411831838, −3.133766635804701, −2.941332866282091, −2.149498277143293, −1.391331969032121, −0.5995888498417419, 0,
0.5995888498417419, 1.391331969032121, 2.149498277143293, 2.941332866282091, 3.133766635804701, 3.891412411831838, 4.473266123292480, 4.934528993182459, 5.563336655535242, 6.070694704842309, 6.423465439139560, 7.007887415607001, 7.358242564132921, 7.779049793226128, 8.721050633628031, 9.010704485871061, 9.519034360985008, 9.989688069841055, 10.29608090459566, 10.83818323625029, 11.27926939170328, 12.14749830690214, 12.20545606232770, 12.50236286304347, 13.22577844520086
