L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s − 11-s + 2·13-s − 15-s − 3·17-s + 6·19-s + 2·21-s − 23-s + 25-s + 27-s + 3·31-s − 33-s − 2·35-s + 37-s + 2·39-s − 5·41-s − 9·43-s − 45-s + 6·47-s − 3·49-s − 3·51-s + 3·53-s + 55-s + 6·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 0.258·15-s − 0.727·17-s + 1.37·19-s + 0.436·21-s − 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.538·31-s − 0.174·33-s − 0.338·35-s + 0.164·37-s + 0.320·39-s − 0.780·41-s − 1.37·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.420·51-s + 0.412·53-s + 0.134·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−12.75574903113505, −12.11481998812694, −11.75528823068630, −11.34758712009079, −10.96614538344483, −10.43523280101454, −9.929322744741478, −9.524050036536290, −8.965709351358949, −8.452454628119283, −8.231530426905301, −7.710618322213680, −7.324429940528616, −6.752489047589910, −6.347684839659998, −5.601563431829429, −5.157816630025312, −4.690159843333177, −4.220698530691638, −3.586497089901205, −3.230046918764784, −2.608997481094693, −2.012683833693540, −1.424958531178700, −0.8778157889008444, 0,
0.8778157889008444, 1.424958531178700, 2.012683833693540, 2.608997481094693, 3.230046918764784, 3.586497089901205, 4.220698530691638, 4.690159843333177, 5.157816630025312, 5.601563431829429, 6.347684839659998, 6.752489047589910, 7.324429940528616, 7.710618322213680, 8.231530426905301, 8.452454628119283, 8.965709351358949, 9.524050036536290, 9.929322744741478, 10.43523280101454, 10.96614538344483, 11.34758712009079, 11.75528823068630, 12.11481998812694, 12.75574903113505