| L(s) = 1 | − 2·3-s + 9-s − 6·11-s − 2·13-s + 17-s − 4·19-s − 5·25-s + 4·27-s − 4·31-s + 12·33-s − 4·37-s + 4·39-s − 6·41-s − 8·43-s − 2·51-s − 6·53-s + 8·57-s + 4·61-s − 8·67-s − 2·73-s + 10·75-s − 8·79-s − 11·81-s + 6·89-s + 8·93-s − 14·97-s − 6·99-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 25-s + 0.769·27-s − 0.718·31-s + 2.08·33-s − 0.657·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.280·51-s − 0.824·53-s + 1.05·57-s + 0.512·61-s − 0.977·67-s − 0.234·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s + 0.635·89-s + 0.829·93-s − 1.42·97-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.89938879345945, −16.10399920112528, −15.95557330708872, −15.06193438769964, −14.79362371269530, −13.79950559221862, −13.33031016271785, −12.74623150851431, −12.19678466193077, −11.73800526495891, −10.95408261021705, −10.64027386824301, −10.08065348148141, −9.512907554292647, −8.443655717530582, −8.119088937246102, −7.300319633094969, −6.742109358879015, −5.950598949559237, −5.393194043090607, −5.021222709598197, −4.253836594933654, −3.235128551131609, −2.471657038647939, −1.593493697187460, 0, 0,
1.593493697187460, 2.471657038647939, 3.235128551131609, 4.253836594933654, 5.021222709598197, 5.393194043090607, 5.950598949559237, 6.742109358879015, 7.300319633094969, 8.119088937246102, 8.443655717530582, 9.512907554292647, 10.08065348148141, 10.64027386824301, 10.95408261021705, 11.73800526495891, 12.19678466193077, 12.74623150851431, 13.33031016271785, 13.79950559221862, 14.79362371269530, 15.06193438769964, 15.95557330708872, 16.10399920112528, 16.89938879345945
