| L(s) = 1 | − 1.58·2-s − 3.81·3-s − 5.50·4-s − 14.4·5-s + 6.02·6-s + 21.3·8-s − 12.4·9-s + 22.8·10-s − 55.1·11-s + 20.9·12-s − 9.01·13-s + 55.0·15-s + 10.2·16-s − 86.5·17-s + 19.6·18-s + 31.5·19-s + 79.4·20-s + 87.1·22-s + 18.3·23-s − 81.4·24-s + 83.5·25-s + 14.2·26-s + 150.·27-s − 248.·29-s − 87.0·30-s − 314.·31-s − 186.·32-s + ⋯ |
| L(s) = 1 | − 0.558·2-s − 0.734·3-s − 0.687·4-s − 1.29·5-s + 0.410·6-s + 0.943·8-s − 0.461·9-s + 0.721·10-s − 1.51·11-s + 0.504·12-s − 0.192·13-s + 0.948·15-s + 0.160·16-s − 1.23·17-s + 0.257·18-s + 0.381·19-s + 0.888·20-s + 0.844·22-s + 0.166·23-s − 0.692·24-s + 0.668·25-s + 0.107·26-s + 1.07·27-s − 1.59·29-s − 0.529·30-s − 1.82·31-s − 1.03·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 1.58T + 8T^{2} \) |
| 3 | \( 1 + 3.81T + 27T^{2} \) |
| 5 | \( 1 + 14.4T + 125T^{2} \) |
| 11 | \( 1 + 55.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.01T + 2.19e3T^{2} \) |
| 17 | \( 1 + 86.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 248.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 314.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 112.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 191.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 672.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 17.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 816.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 308.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 876.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 102.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 258.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 790.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 97.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 264.T + 9.12e5T^{2} \) |
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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
−8.289132951392743597014539082724, −7.53315003777353607617509644784, −7.02475023486540833802156699987, −5.58414499789975627740973245386, −5.17677555079072626694381710981, −4.26391914662793960348851308137, −3.46400349953954610353112167185, −2.18540820226419636109647263865, −0.54178586998790196465200337168, 0,
0.54178586998790196465200337168, 2.18540820226419636109647263865, 3.46400349953954610353112167185, 4.26391914662793960348851308137, 5.17677555079072626694381710981, 5.58414499789975627740973245386, 7.02475023486540833802156699987, 7.53315003777353607617509644784, 8.289132951392743597014539082724
