| L(s) = 1 | − 3·3-s + 5-s + 6·9-s − 2·11-s + 6·13-s − 3·15-s + 2·17-s + 9·23-s + 25-s − 9·27-s − 3·29-s − 2·31-s + 6·33-s − 8·37-s − 18·39-s + 5·41-s + 43-s + 6·45-s − 8·47-s − 6·51-s − 4·53-s − 2·55-s − 8·59-s − 7·61-s + 6·65-s − 3·67-s − 27·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s − 0.603·11-s + 1.66·13-s − 0.774·15-s + 0.485·17-s + 1.87·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s − 0.359·31-s + 1.04·33-s − 1.31·37-s − 2.88·39-s + 0.780·41-s + 0.152·43-s + 0.894·45-s − 1.16·47-s − 0.840·51-s − 0.549·53-s − 0.269·55-s − 1.04·59-s − 0.896·61-s + 0.744·65-s − 0.366·67-s − 3.25·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 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
−16.53924276097805, −15.77857248371115, −15.45929705373703, −14.69731542667177, −13.90533249339385, −13.17456307650120, −13.01547542774781, −12.31966425352184, −11.72625330877742, −10.98154620917147, −10.82970110505596, −10.40383093810963, −9.462837645016167, −9.045919067362497, −8.145357139228173, −7.440537656264192, −6.682732071460288, −6.327731468127015, −5.553588370459019, −5.286907398886290, −4.584172422430412, −3.694621923759365, −2.935600884949372, −1.587870410305655, −1.111994902792796, 0,
1.111994902792796, 1.587870410305655, 2.935600884949372, 3.694621923759365, 4.584172422430412, 5.286907398886290, 5.553588370459019, 6.327731468127015, 6.682732071460288, 7.440537656264192, 8.145357139228173, 9.045919067362497, 9.462837645016167, 10.40383093810963, 10.82970110505596, 10.98154620917147, 11.72625330877742, 12.31966425352184, 13.01547542774781, 13.17456307650120, 13.90533249339385, 14.69731542667177, 15.45929705373703, 15.77857248371115, 16.53924276097805
