| L(s) = 1 | − 2·3-s + 5-s + 9-s − 4·11-s − 2·13-s − 2·15-s − 17-s − 19-s + 23-s − 4·25-s + 4·27-s + 6·29-s + 4·31-s + 8·33-s + 3·37-s + 4·39-s − 10·41-s + 43-s + 45-s + 4·47-s + 2·51-s − 6·53-s − 4·55-s + 2·57-s − 59-s + 10·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.229·19-s + 0.208·23-s − 4/5·25-s + 0.769·27-s + 1.11·29-s + 0.718·31-s + 1.39·33-s + 0.493·37-s + 0.640·39-s − 1.56·41-s + 0.152·43-s + 0.149·45-s + 0.583·47-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.264·57-s − 0.130·59-s + 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 53312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−14.75069565024614, −14.03276448985293, −13.70985382675716, −13.02196679100362, −12.71026078593208, −11.99813532635673, −11.73164521107210, −11.09185732907609, −10.61063061266641, −10.03231735959499, −9.920687876751364, −8.973304775144183, −8.420179954633661, −7.865162797267148, −7.274733323813025, −6.486214539149495, −6.307097204756906, −5.486506945645918, −5.156980707773537, −4.684827733206032, −3.957059168441883, −2.923634184996439, −2.532868362122679, −1.692006856643341, −0.7263311737616488, 0,
0.7263311737616488, 1.692006856643341, 2.532868362122679, 2.923634184996439, 3.957059168441883, 4.684827733206032, 5.156980707773537, 5.486506945645918, 6.307097204756906, 6.486214539149495, 7.274733323813025, 7.865162797267148, 8.420179954633661, 8.973304775144183, 9.920687876751364, 10.03231735959499, 10.61063061266641, 11.09185732907609, 11.73164521107210, 11.99813532635673, 12.71026078593208, 13.02196679100362, 13.70985382675716, 14.03276448985293, 14.75069565024614
