| L(s) = 1 | − 5·13-s − 17-s − 19-s − 6·23-s − 5·25-s + 6·29-s + 5·31-s − 7·37-s − 6·41-s + 43-s + 6·47-s − 6·53-s + 10·61-s − 5·67-s − 6·71-s + 73-s + 79-s + 6·83-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.38·13-s − 0.242·17-s − 0.229·19-s − 1.25·23-s − 25-s + 1.11·29-s + 0.898·31-s − 1.15·37-s − 0.937·41-s + 0.152·43-s + 0.875·47-s − 0.824·53-s + 1.28·61-s − 0.610·67-s − 0.712·71-s + 0.117·73-s + 0.112·79-s + 0.658·83-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6840613356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6840613356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−13.63132977540401, −13.13870030537707, −12.42666326290183, −12.05287526554428, −11.87113129429431, −11.21368333835274, −10.43069349261912, −10.20480136963220, −9.779982496922967, −9.198854230846640, −8.632937470112733, −8.032318510950471, −7.774191787413294, −7.022248469487476, −6.641305439075312, −6.082036896011394, −5.419180134424858, −4.973321915384123, −4.338178773636737, −3.923458108557542, −3.134737515756740, −2.492269208499065, −2.052298089960096, −1.282338382371784, −0.2487168106291979,
0.2487168106291979, 1.282338382371784, 2.052298089960096, 2.492269208499065, 3.134737515756740, 3.923458108557542, 4.338178773636737, 4.973321915384123, 5.419180134424858, 6.082036896011394, 6.641305439075312, 7.022248469487476, 7.774191787413294, 8.032318510950471, 8.632937470112733, 9.198854230846640, 9.779982496922967, 10.20480136963220, 10.43069349261912, 11.21368333835274, 11.87113129429431, 12.05287526554428, 12.42666326290183, 13.13870030537707, 13.63132977540401
