| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 5·11-s + 15-s + 17-s + 19-s + 2·21-s − 8·23-s + 25-s − 27-s + 3·29-s + 8·31-s + 5·33-s + 2·35-s − 10·37-s + 3·41-s − 9·43-s − 45-s + 2·47-s − 3·49-s − 51-s − 4·53-s + 5·55-s − 57-s − 14·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 1.50·11-s + 0.258·15-s + 0.242·17-s + 0.229·19-s + 0.436·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 1.43·31-s + 0.870·33-s + 0.338·35-s − 1.64·37-s + 0.468·41-s − 1.37·43-s − 0.149·45-s + 0.291·47-s − 3/7·49-s − 0.140·51-s − 0.549·53-s + 0.674·55-s − 0.132·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−12.73531329922562, −12.18667666584398, −11.94990766689354, −11.65084604783502, −10.72100104240267, −10.54741183924422, −10.25480529741514, −9.663120378867069, −9.325128618559876, −8.478775103498080, −8.174013344382629, −7.741142316839359, −7.304002218338199, −6.676297566490484, −6.219650729542626, −5.898220760476860, −5.223513454265963, −4.803608287718941, −4.400982544513571, −3.639460466507855, −3.179169732017060, −2.753246736478786, −2.014545098596332, −1.389908818521313, −0.4558056415328583, 0,
0.4558056415328583, 1.389908818521313, 2.014545098596332, 2.753246736478786, 3.179169732017060, 3.639460466507855, 4.400982544513571, 4.803608287718941, 5.223513454265963, 5.898220760476860, 6.219650729542626, 6.676297566490484, 7.304002218338199, 7.741142316839359, 8.174013344382629, 8.478775103498080, 9.325128618559876, 9.663120378867069, 10.25480529741514, 10.54741183924422, 10.72100104240267, 11.65084604783502, 11.94990766689354, 12.18667666584398, 12.73531329922562
