| L(s) = 1 | + 3-s + 2.15·5-s + 2.43·7-s + 9-s − 5.31·11-s − 3.43·13-s + 2.15·15-s − 2.80·17-s − 1.25·19-s + 2.43·21-s − 5.03·23-s − 0.344·25-s + 27-s − 2.59·29-s − 10.9·31-s − 5.31·33-s + 5.24·35-s + 10.3·37-s − 3.43·39-s + 10.9·41-s − 3.96·43-s + 2.15·45-s − 3.18·47-s − 1.09·49-s − 2.80·51-s + 0.771·53-s − 11.4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.964·5-s + 0.918·7-s + 0.333·9-s − 1.60·11-s − 0.951·13-s + 0.557·15-s − 0.680·17-s − 0.288·19-s + 0.530·21-s − 1.04·23-s − 0.0689·25-s + 0.192·27-s − 0.482·29-s − 1.96·31-s − 0.924·33-s + 0.886·35-s + 1.70·37-s − 0.549·39-s + 1.71·41-s − 0.604·43-s + 0.321·45-s − 0.465·47-s − 0.155·49-s − 0.392·51-s + 0.106·53-s − 1.54·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 5 | \( 1 - 2.15T + 5T^{2} \) |
| 7 | \( 1 - 2.43T + 7T^{2} \) |
| 11 | \( 1 + 5.31T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 2.80T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 + 5.03T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.96T + 43T^{2} \) |
| 47 | \( 1 + 3.18T + 47T^{2} \) |
| 53 | \( 1 - 0.771T + 53T^{2} \) |
| 59 | \( 1 - 0.581T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 6.69T + 67T^{2} \) |
| 71 | \( 1 - 0.567T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 2.52T + 89T^{2} \) |
| 97 | \( 1 - 8.23T + 97T^{2} \) |
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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
−7.57646921826220830111034939647, −7.42594621716746255553575721221, −6.08820724424018622961395124074, −5.60923944958502028608805197113, −4.80010958046673342034760200669, −4.20865398434442286693627351808, −2.92154806495854280854352506656, −2.22122749201396777042160357703, −1.75544337995236797208101846138, 0,
1.75544337995236797208101846138, 2.22122749201396777042160357703, 2.92154806495854280854352506656, 4.20865398434442286693627351808, 4.80010958046673342034760200669, 5.60923944958502028608805197113, 6.08820724424018622961395124074, 7.42594621716746255553575721221, 7.57646921826220830111034939647
