| L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·11-s − 3·13-s − 2·15-s + 8·17-s + 19-s − 8·23-s − 25-s + 27-s + 4·29-s − 3·31-s − 2·33-s − 37-s − 3·39-s + 6·41-s − 11·43-s − 2·45-s − 6·47-s + 8·51-s − 12·53-s + 4·55-s + 57-s − 4·59-s − 6·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.832·13-s − 0.516·15-s + 1.94·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.538·31-s − 0.348·33-s − 0.164·37-s − 0.480·39-s + 0.937·41-s − 1.67·43-s − 0.298·45-s − 0.875·47-s + 1.12·51-s − 1.64·53-s + 0.539·55-s + 0.132·57-s − 0.520·59-s − 0.768·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.304389845549705419721349874798, −7.80305217239486941411038316758, −7.48792818245316224281758841542, −6.29726913873598404702318801092, −5.34353355016647543832466490803, −4.49676339846623379295899616014, −3.55326335635197764003572865473, −2.90196711465655366821264757704, −1.63793208380212130696209039180, 0,
1.63793208380212130696209039180, 2.90196711465655366821264757704, 3.55326335635197764003572865473, 4.49676339846623379295899616014, 5.34353355016647543832466490803, 6.29726913873598404702318801092, 7.48792818245316224281758841542, 7.80305217239486941411038316758, 8.304389845549705419721349874798
