| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 11-s + 13-s + 16-s + 4·17-s − 2·19-s − 20-s − 22-s + 6·23-s + 25-s + 26-s + 2·29-s − 10·31-s + 32-s + 4·34-s − 4·37-s − 2·38-s − 40-s + 2·41-s + 8·43-s − 44-s + 6·46-s + 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.458·19-s − 0.223·20-s − 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.371·29-s − 1.79·31-s + 0.176·32-s + 0.685·34-s − 0.657·37-s − 0.324·38-s − 0.158·40-s + 0.312·41-s + 1.21·43-s − 0.150·44-s + 0.884·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182576005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182576005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.08145043621067, −15.77638264664992, −14.90350054883016, −14.75236896711106, −14.03913349366848, −13.44315230878989, −12.76085294085319, −12.46733373860006, −11.83972261240981, −11.00474064965826, −10.84729376341462, −10.08269935025186, −9.247167720983415, −8.700521573504086, −7.935717547857965, −7.314947615101984, −6.891498852175773, −5.931549785185951, −5.470546668346106, −4.769844640064453, −4.014989483728844, −3.394917700191567, −2.724495564821650, −1.754360424829762, −0.7252681354111452,
0.7252681354111452, 1.754360424829762, 2.724495564821650, 3.394917700191567, 4.014989483728844, 4.769844640064453, 5.470546668346106, 5.931549785185951, 6.891498852175773, 7.314947615101984, 7.935717547857965, 8.700521573504086, 9.247167720983415, 10.08269935025186, 10.84729376341462, 11.00474064965826, 11.83972261240981, 12.46733373860006, 12.76085294085319, 13.44315230878989, 14.03913349366848, 14.75236896711106, 14.90350054883016, 15.77638264664992, 16.08145043621067
