| L(s) = 1 | − 1.50·3-s − 37.8·5-s + 124.·7-s − 240.·9-s − 590.·11-s + 434.·13-s + 56.8·15-s + 1.92e3·17-s − 654.·19-s − 187.·21-s − 2.80e3·23-s − 1.69e3·25-s + 726.·27-s − 1.45e3·29-s − 4.41e3·31-s + 886.·33-s − 4.71e3·35-s + 3.75e3·37-s − 652.·39-s + 1.97e3·41-s − 1.84e3·43-s + 9.10e3·45-s − 2.20e3·47-s − 1.24e3·49-s − 2.89e3·51-s + 2.49e4·53-s + 2.23e4·55-s + ⋯ |
| L(s) = 1 | − 0.0963·3-s − 0.676·5-s + 0.962·7-s − 0.990·9-s − 1.47·11-s + 0.713·13-s + 0.0651·15-s + 1.61·17-s − 0.415·19-s − 0.0926·21-s − 1.10·23-s − 0.542·25-s + 0.191·27-s − 0.321·29-s − 0.826·31-s + 0.141·33-s − 0.651·35-s + 0.450·37-s − 0.0687·39-s + 0.183·41-s − 0.152·43-s + 0.670·45-s − 0.145·47-s − 0.0740·49-s − 0.155·51-s + 1.22·53-s + 0.994·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.289018401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.289018401\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + 1.84e3T \) |
| good | 3 | \( 1 + 1.50T + 243T^{2} \) |
| 5 | \( 1 + 37.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 590.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 434.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 654.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.75e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.97e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.20e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.49e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.10e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.27e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.84e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.03e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.88e4T + 8.58e9T^{2} \) |
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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
−9.835592837797585761836229548444, −8.425580700353786353468997311049, −8.111538405392292958214611972664, −7.38790437952445591287608710556, −5.80551536560676146122987281622, −5.38988169336153700926978802928, −4.14876550821081339768009448861, −3.14168865226634523649233017318, −1.95260316044086946412696515882, −0.52393223153185918478239787029,
0.52393223153185918478239787029, 1.95260316044086946412696515882, 3.14168865226634523649233017318, 4.14876550821081339768009448861, 5.38988169336153700926978802928, 5.80551536560676146122987281622, 7.38790437952445591287608710556, 8.111538405392292958214611972664, 8.425580700353786353468997311049, 9.835592837797585761836229548444
