| L(s) = 1 | − 5-s − 3·7-s + 5·13-s + 3·17-s + 19-s + 6·23-s + 25-s + 9·29-s + 4·31-s + 3·35-s + 3·37-s − 8·41-s + 4·43-s + 2·47-s + 2·49-s − 2·53-s + 6·59-s + 6·61-s − 5·65-s + 14·67-s − 3·71-s − 4·73-s + 6·79-s + 13·83-s − 3·85-s − 15·91-s − 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.38·13-s + 0.727·17-s + 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.67·29-s + 0.718·31-s + 0.507·35-s + 0.493·37-s − 1.24·41-s + 0.609·43-s + 0.291·47-s + 2/7·49-s − 0.274·53-s + 0.781·59-s + 0.768·61-s − 0.620·65-s + 1.71·67-s − 0.356·71-s − 0.468·73-s + 0.675·79-s + 1.42·83-s − 0.325·85-s − 1.57·91-s − 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.535351061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.535351061\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.68366793245112, −14.12640096771013, −13.54589044252262, −13.17815148677667, −12.63200273817187, −12.14592465541240, −11.57054560604950, −11.08461085705511, −10.44259072795084, −10.00777312640770, −9.476863614058713, −8.727921364848461, −8.474509052692786, −7.779445838944915, −7.114706778658145, −6.460526022702186, −6.286204806359994, −5.417734591149102, −4.836155100610401, −4.075891962089682, −3.326744754141560, −3.191872934573927, −2.282285347120080, −1.071843788199583, −0.7118666875027953,
0.7118666875027953, 1.071843788199583, 2.282285347120080, 3.191872934573927, 3.326744754141560, 4.075891962089682, 4.836155100610401, 5.417734591149102, 6.286204806359994, 6.460526022702186, 7.114706778658145, 7.779445838944915, 8.474509052692786, 8.727921364848461, 9.476863614058713, 10.00777312640770, 10.44259072795084, 11.08461085705511, 11.57054560604950, 12.14592465541240, 12.63200273817187, 13.17815148677667, 13.54589044252262, 14.12640096771013, 14.68366793245112
