| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s + 13-s + 4·14-s + 15-s + 16-s − 4·17-s + 18-s + 2·19-s + 20-s + 4·21-s + 6·23-s + 24-s + 25-s + 26-s + 27-s + 4·28-s + 6·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.872·21-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-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
| \(L(1)\) |
\(\approx\) |
\(7.869397415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.869397415\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−14.38380071349103, −14.11613709198338, −13.69712919987177, −13.17537837621478, −12.68787634441714, −12.00458000104575, −11.51053933926695, −11.02981816722772, −10.59132677319461, −9.985308779326993, −9.279371678818663, −8.545310574653821, −8.461502908417099, −7.690102504591365, −6.971973448923222, −6.726145880084497, −5.854090684295744, −5.092441563321170, −4.890182335722327, −4.295194807856082, −3.433110244522607, −2.954109548777008, −2.019673028245822, −1.737917306874012, −0.8727565805402025,
0.8727565805402025, 1.737917306874012, 2.019673028245822, 2.954109548777008, 3.433110244522607, 4.295194807856082, 4.890182335722327, 5.092441563321170, 5.854090684295744, 6.726145880084497, 6.971973448923222, 7.690102504591365, 8.461502908417099, 8.545310574653821, 9.279371678818663, 9.985308779326993, 10.59132677319461, 11.02981816722772, 11.51053933926695, 12.00458000104575, 12.68787634441714, 13.17537837621478, 13.69712919987177, 14.11613709198338, 14.38380071349103
