| L(s) = 1 | + 2-s + 4-s + 5-s + 7-s + 8-s + 10-s + 4·11-s − 2·13-s + 14-s + 16-s + 17-s − 4·19-s + 20-s + 4·22-s − 8·23-s + 25-s − 2·26-s + 28-s − 6·29-s − 31-s + 32-s + 34-s + 35-s − 2·37-s − 4·38-s + 40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s + 0.353·8-s + 0.316·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s + 0.223·20-s + 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.392·26-s + 0.188·28-s − 1.11·29-s − 0.179·31-s + 0.176·32-s + 0.171·34-s + 0.169·35-s − 0.328·37-s − 0.648·38-s + 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.356354952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.356354952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.59002593026896, −12.02556036373247, −11.95370313763771, −11.28569099874544, −10.88718151287492, −10.32287792898240, −9.955481084237297, −9.425875076222978, −8.962440102587565, −8.473858963749353, −7.932054671660936, −7.354816646822239, −7.043506694345519, −6.340519893037189, −6.017787864980794, −5.655974264925660, −5.002806119512825, −4.400864243242397, −4.146777290521653, −3.556251072572242, −2.983923471229090, −2.267599116425896, −1.717219234791073, −1.508135304923061, −0.3849710781798285,
0.3849710781798285, 1.508135304923061, 1.717219234791073, 2.267599116425896, 2.983923471229090, 3.556251072572242, 4.146777290521653, 4.400864243242397, 5.002806119512825, 5.655974264925660, 6.017787864980794, 6.340519893037189, 7.043506694345519, 7.354816646822239, 7.932054671660936, 8.473858963749353, 8.962440102587565, 9.425875076222978, 9.955481084237297, 10.32287792898240, 10.88718151287492, 11.28569099874544, 11.95370313763771, 12.02556036373247, 12.59002593026896
