| L(s) = 1 | − 3-s − 4.10·5-s + 7-s + 9-s − 2.67·11-s − 3.02·13-s + 4.10·15-s − 5.12·17-s + 2.78·19-s − 21-s + 7.12·23-s + 11.8·25-s − 27-s + 8.83·29-s + 1.42·31-s + 2.67·33-s − 4.10·35-s + 1.42·37-s + 3.02·39-s − 5.12·41-s − 2.39·43-s − 4.10·45-s + 9.56·47-s + 49-s + 5.12·51-s − 2.78·53-s + 10.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.83·5-s + 0.377·7-s + 0.333·9-s − 0.807·11-s − 0.838·13-s + 1.05·15-s − 1.24·17-s + 0.637·19-s − 0.218·21-s + 1.48·23-s + 2.36·25-s − 0.192·27-s + 1.63·29-s + 0.255·31-s + 0.466·33-s − 0.693·35-s + 0.234·37-s + 0.484·39-s − 0.800·41-s − 0.365·43-s − 0.611·45-s + 1.39·47-s + 0.142·49-s + 0.717·51-s − 0.381·53-s + 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4.10T + 5T^{2} \) |
| 11 | \( 1 + 2.67T + 11T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 2.78T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 8.83T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 + 5.12T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 5.17T + 61T^{2} \) |
| 67 | \( 1 + 0.244T + 67T^{2} \) |
| 71 | \( 1 + 4.27T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 + 6.25T + 79T^{2} \) |
| 83 | \( 1 + 9.35T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−7.73525556676813909366262227331, −7.13551161482870853532793760031, −6.69046904098119812286592217586, −5.42103868544098599096451669320, −4.67181693767290745726557612868, −4.44931882613903435220505023683, −3.27881814153360258795933919503, −2.56033683328850313102071897212, −0.968094808643267249743844037179, 0,
0.968094808643267249743844037179, 2.56033683328850313102071897212, 3.27881814153360258795933919503, 4.44931882613903435220505023683, 4.67181693767290745726557612868, 5.42103868544098599096451669320, 6.69046904098119812286592217586, 7.13551161482870853532793760031, 7.73525556676813909366262227331
