| L(s) = 1 | + 3-s + 1.23·5-s + 9-s + 11-s − 13-s + 1.23·15-s − 5.23·17-s − 2.47·19-s − 4·23-s − 3.47·25-s + 27-s + 1.23·29-s − 3.23·31-s + 33-s + 0.472·37-s − 39-s + 12.4·41-s − 5.70·43-s + 1.23·45-s − 4·47-s − 7·49-s − 5.23·51-s − 2·53-s + 1.23·55-s − 2.47·57-s − 10.4·59-s − 4.47·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.552·5-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.319·15-s − 1.26·17-s − 0.567·19-s − 0.834·23-s − 0.694·25-s + 0.192·27-s + 0.229·29-s − 0.581·31-s + 0.174·33-s + 0.0776·37-s − 0.160·39-s + 1.94·41-s − 0.870·43-s + 0.184·45-s − 0.583·47-s − 49-s − 0.733·51-s − 0.274·53-s + 0.166·55-s − 0.327·57-s − 1.36·59-s − 0.572·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.23T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 + 3.23T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 0.763T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−7.77432528993576502899890469810, −6.82221858406693904008261199301, −6.31614143362224937156444773543, −5.57812386120543754419872677516, −4.55305562941864657064600616535, −4.08509574251541827073450200689, −3.05653036017541197730231224156, −2.21489667621299387035845749229, −1.59744671188861808410776280296, 0,
1.59744671188861808410776280296, 2.21489667621299387035845749229, 3.05653036017541197730231224156, 4.08509574251541827073450200689, 4.55305562941864657064600616535, 5.57812386120543754419872677516, 6.31614143362224937156444773543, 6.82221858406693904008261199301, 7.77432528993576502899890469810
