| L(s) = 1 | + 0.836·2-s − 6.41·3-s − 7.29·4-s + 1.43·5-s − 5.36·6-s − 12.8·8-s + 14.1·9-s + 1.19·10-s + 54.3·11-s + 46.8·12-s + 67.1·13-s − 9.18·15-s + 47.6·16-s + 18.9·17-s + 11.8·18-s − 58.6·19-s − 10.4·20-s + 45.4·22-s + 19.2·23-s + 82.1·24-s − 122.·25-s + 56.2·26-s + 82.3·27-s + 228.·29-s − 7.68·30-s + 165.·31-s + 142.·32-s + ⋯ |
| L(s) = 1 | + 0.295·2-s − 1.23·3-s − 0.912·4-s + 0.128·5-s − 0.365·6-s − 0.565·8-s + 0.524·9-s + 0.0378·10-s + 1.49·11-s + 1.12·12-s + 1.43·13-s − 0.158·15-s + 0.744·16-s + 0.269·17-s + 0.155·18-s − 0.707·19-s − 0.116·20-s + 0.440·22-s + 0.174·23-s + 0.698·24-s − 0.983·25-s + 0.424·26-s + 0.587·27-s + 1.46·29-s − 0.0467·30-s + 0.957·31-s + 0.786·32-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
| \(L(2)\) |
\(\approx\) |
\(1.541816185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541816185\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 0.836T + 8T^{2} \) |
| 3 | \( 1 + 6.41T + 27T^{2} \) |
| 5 | \( 1 - 1.43T + 125T^{2} \) |
| 11 | \( 1 - 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 58.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 228.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 228.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 531.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 167.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 453.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 619.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 396.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 981.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−8.724061389572754181478103746901, −7.956473988483423588300368516088, −6.60281664578451494586018558176, −6.12081076449938858724498362910, −5.68613919003363181137917303902, −4.43866564630845204425563483622, −4.20521088732143562076791534005, −3.05388814970155573326227719449, −1.31357598610042430663043297206, −0.66218942088693057247373099291,
0.66218942088693057247373099291, 1.31357598610042430663043297206, 3.05388814970155573326227719449, 4.20521088732143562076791534005, 4.43866564630845204425563483622, 5.68613919003363181137917303902, 6.12081076449938858724498362910, 6.60281664578451494586018558176, 7.956473988483423588300368516088, 8.724061389572754181478103746901
