| L(s) = 1 | + 2·3-s + 5-s − 2·7-s + 9-s − 11-s − 6·13-s + 2·15-s − 17-s + 4·19-s − 4·21-s − 2·23-s + 25-s − 4·27-s + 2·29-s − 2·33-s − 2·35-s − 2·37-s − 12·39-s + 6·41-s + 10·43-s + 45-s + 6·47-s − 3·49-s − 2·51-s + 6·53-s − 55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.242·17-s + 0.917·19-s − 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.348·33-s − 0.338·35-s − 0.328·37-s − 1.92·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s − 0.134·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.59327266792030, −13.95432872685132, −13.76466060584394, −13.12473887727963, −12.61589503967985, −12.20722810113913, −11.64378240690215, −10.88785249403028, −10.27268077603025, −9.822969083926362, −9.347237992089412, −9.140629736612872, −8.356068306032293, −7.801172704818063, −7.361775431107925, −6.874666985619549, −6.150522112853931, −5.481982356734961, −5.062708006083234, −4.175199361463737, −3.692275832866747, −2.810744398419729, −2.592942239531390, −2.076581264280522, −0.9739127312507911, 0,
0.9739127312507911, 2.076581264280522, 2.592942239531390, 2.810744398419729, 3.692275832866747, 4.175199361463737, 5.062708006083234, 5.481982356734961, 6.150522112853931, 6.874666985619549, 7.361775431107925, 7.801172704818063, 8.356068306032293, 9.140629736612872, 9.347237992089412, 9.822969083926362, 10.27268077603025, 10.88785249403028, 11.64378240690215, 12.20722810113913, 12.61589503967985, 13.12473887727963, 13.76466060584394, 13.95432872685132, 14.59327266792030
