| L(s) = 1 | + 2·3-s + 9-s − 4·11-s + 2·13-s − 2·19-s + 4·23-s − 4·27-s − 10·29-s + 4·31-s − 8·33-s − 2·37-s + 4·39-s − 12·41-s − 4·43-s − 4·47-s + 2·53-s − 4·57-s − 10·59-s − 6·61-s + 4·67-s + 8·69-s − 12·71-s + 4·73-s − 4·79-s − 11·81-s + 14·83-s − 20·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 0.458·19-s + 0.834·23-s − 0.769·27-s − 1.85·29-s + 0.718·31-s − 1.39·33-s − 0.328·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s − 0.583·47-s + 0.274·53-s − 0.529·57-s − 1.30·59-s − 0.768·61-s + 0.488·67-s + 0.963·69-s − 1.42·71-s + 0.468·73-s − 0.450·79-s − 1.22·81-s + 1.53·83-s − 2.14·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.968452201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968452201\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.78092790322093, −13.51097871132866, −13.29664852531159, −12.71110761453987, −12.08278917639117, −11.50703972759241, −10.83398084789027, −10.63232613959893, −9.856773681765885, −9.450233850807781, −8.845954044486824, −8.450372023877719, −7.989127787141158, −7.548114356594582, −6.940934851549810, −6.331084324707232, −5.602926686204016, −5.163118308619972, −4.492948186298304, −3.774644150222712, −3.143193240488118, −2.917821748413713, −1.962660026154292, −1.662467980360694, −0.3935858673403982,
0.3935858673403982, 1.662467980360694, 1.962660026154292, 2.917821748413713, 3.143193240488118, 3.774644150222712, 4.492948186298304, 5.163118308619972, 5.602926686204016, 6.331084324707232, 6.940934851549810, 7.548114356594582, 7.989127787141158, 8.450372023877719, 8.845954044486824, 9.450233850807781, 9.856773681765885, 10.63232613959893, 10.83398084789027, 11.50703972759241, 12.08278917639117, 12.71110761453987, 13.29664852531159, 13.51097871132866, 13.78092790322093
