| L(s) = 1 | − 3-s + 5-s − 2·9-s − 3·11-s − 6·13-s − 15-s − 5·17-s − 19-s + 7·23-s − 4·25-s + 5·27-s − 2·29-s + 3·33-s − 3·37-s + 6·39-s + 2·41-s + 4·43-s − 2·45-s − 5·47-s + 5·51-s + 53-s − 3·55-s + 57-s − 15·59-s − 5·61-s − 6·65-s − 9·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 2/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s − 1.21·17-s − 0.229·19-s + 1.45·23-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.522·33-s − 0.493·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s − 0.298·45-s − 0.729·47-s + 0.700·51-s + 0.137·53-s − 0.404·55-s + 0.132·57-s − 1.95·59-s − 0.640·61-s − 0.744·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 376712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−12.79668617971088, −12.18207874970279, −11.84263830028608, −11.32279340180991, −10.86806650840116, −10.51313678681271, −10.17976772937540, −9.339003464964792, −9.254759196672517, −8.774944309441324, −8.022317593828428, −7.633256813018668, −7.258578106608990, −6.541492667889566, −6.320191080567452, −5.657063463386505, −5.215873305625249, −4.809058862994397, −4.545604164314594, −3.657214281755091, −2.965402540424148, −2.556752542216573, −2.158281250465194, −1.451626764495031, −0.4624197006319403, 0,
0.4624197006319403, 1.451626764495031, 2.158281250465194, 2.556752542216573, 2.965402540424148, 3.657214281755091, 4.545604164314594, 4.809058862994397, 5.215873305625249, 5.657063463386505, 6.320191080567452, 6.541492667889566, 7.258578106608990, 7.633256813018668, 8.022317593828428, 8.774944309441324, 9.254759196672517, 9.339003464964792, 10.17976772937540, 10.51313678681271, 10.86806650840116, 11.32279340180991, 11.84263830028608, 12.18207874970279, 12.79668617971088
