| L(s) = 1 | − 0.709·2-s − 0.497·4-s − 0.241·5-s + 1.06·8-s + 9-s + 0.170·10-s − 0.255·16-s − 17-s − 0.709·18-s + 1.13·19-s + 0.119·20-s + 0.709·23-s − 0.941·25-s − 0.880·32-s + 0.709·34-s − 0.497·36-s + 1.49·37-s − 0.805·38-s − 0.255·40-s − 1.13·41-s − 0.241·45-s − 0.502·46-s + 0.241·47-s + 49-s + 0.667·50-s + 0.880·64-s + 0.497·68-s + ⋯ |
| L(s) = 1 | − 0.709·2-s − 0.497·4-s − 0.241·5-s + 1.06·8-s + 9-s + 0.170·10-s − 0.255·16-s − 17-s − 0.709·18-s + 1.13·19-s + 0.119·20-s + 0.709·23-s − 0.941·25-s − 0.880·32-s + 0.709·34-s − 0.497·36-s + 1.49·37-s − 0.805·38-s − 0.255·40-s − 1.13·41-s − 0.241·45-s − 0.502·46-s + 0.241·47-s + 49-s + 0.667·50-s + 0.880·64-s + 0.497·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2839 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2839 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7504280355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7504280355\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 2 | \( 1 + 0.709T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 0.241T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.13T + T^{2} \) |
| 23 | \( 1 - 0.709T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.49T + T^{2} \) |
| 41 | \( 1 + 1.13T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 0.241T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.94T + T^{2} \) |
| 73 | \( 1 + 1.77T + T^{2} \) |
| 79 | \( 1 + 1.13T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.77T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.017439856142200047400667677582, −8.299744284359451611487291219048, −7.46547581241142866405548666469, −7.08685366097817923502240887132, −5.95476829235663568301723567447, −4.90089038632400998320130252024, −4.33330412911843089122772274670, −3.45073772382911449250393777982, −2.05043103752301288068819645013, −0.925142473745484495893715475996,
0.925142473745484495893715475996, 2.05043103752301288068819645013, 3.45073772382911449250393777982, 4.33330412911843089122772274670, 4.90089038632400998320130252024, 5.95476829235663568301723567447, 7.08685366097817923502240887132, 7.46547581241142866405548666469, 8.299744284359451611487291219048, 9.017439856142200047400667677582
