| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s − 13-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s − 24-s + 25-s + 26-s + 27-s + 6·29-s + 30-s + 4·31-s − 32-s + 6·34-s + 36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.718·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.06025886970500, −14.33672991356087, −13.94431358357388, −13.20631231755135, −12.86662246260773, −12.15245401068622, −11.76907025658529, −11.11487144693206, −10.58080687250263, −10.22153345343166, −9.544223076322124, −8.904358947484653, −8.590451182117326, −8.187095519620129, −7.426468357886961, −7.023016485253136, −6.459951147659481, −5.896361159323048, −4.922668482889382, −4.351491488416268, −3.900524334430116, −2.907070066307270, −2.506857309988162, −1.820502503498783, −0.8829622252669570, 0,
0.8829622252669570, 1.820502503498783, 2.506857309988162, 2.907070066307270, 3.900524334430116, 4.351491488416268, 4.922668482889382, 5.896361159323048, 6.459951147659481, 7.023016485253136, 7.426468357886961, 8.187095519620129, 8.590451182117326, 8.904358947484653, 9.544223076322124, 10.22153345343166, 10.58080687250263, 11.11487144693206, 11.76907025658529, 12.15245401068622, 12.86662246260773, 13.20631231755135, 13.94431358357388, 14.33672991356087, 15.06025886970500
