| L(s) = 1 | − 3·3-s + 6·9-s + 2·11-s + 6·13-s − 2·17-s − 9·23-s − 9·27-s + 3·29-s − 2·31-s − 6·33-s − 8·37-s − 18·39-s + 5·41-s + 43-s + 8·47-s + 6·51-s − 4·53-s + 8·59-s + 7·61-s − 3·67-s + 27·69-s − 8·71-s − 14·73-s − 4·79-s + 9·81-s − 83-s − 9·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 0.603·11-s + 1.66·13-s − 0.485·17-s − 1.87·23-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 + 1.16·47-s + 0.840·51-s − 0.549·53-s + 1.04·59-s + 0.896·61-s − 0.366·67-s + 3.25·69-s − 0.949·71-s − 1.63·73-s − 0.450·79-s + 81-s − 0.109·83-s − 0.964·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059249978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059249978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 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
−15.94379705439641, −15.53239679853367, −14.51851326537154, −14.03058887131026, −13.34071988436142, −12.90723944310900, −12.13150133839951, −11.82539093152319, −11.36400451686587, −10.67503660186602, −10.43206581220756, −9.746724759474576, −8.903205473038941, −8.460605877402554, −7.560212885927335, −6.906916656756699, −6.313542486013562, −5.912326382144491, −5.489764440619392, −4.495141465937421, −4.109927731090198, −3.420512708782118, −2.059851193423438, −1.334914681370606, −0.5180587815676894,
0.5180587815676894, 1.334914681370606, 2.059851193423438, 3.420512708782118, 4.109927731090198, 4.495141465937421, 5.489764440619392, 5.912326382144491, 6.313542486013562, 6.906916656756699, 7.560212885927335, 8.460605877402554, 8.903205473038941, 9.746724759474576, 10.43206581220756, 10.67503660186602, 11.36400451686587, 11.82539093152319, 12.13150133839951, 12.90723944310900, 13.34071988436142, 14.03058887131026, 14.51851326537154, 15.53239679853367, 15.94379705439641
