| L(s) = 1 | − 3.04·3-s − 5-s + 0.574·7-s + 6.26·9-s − 2.57·11-s − 0.468·13-s + 3.04·15-s − 4.08·17-s − 19-s − 1.74·21-s − 1.51·23-s + 25-s − 9.92·27-s − 4.08·29-s + 9.92·31-s + 7.83·33-s − 0.574·35-s − 8.30·37-s + 1.42·39-s − 1.83·41-s + 0.574·43-s − 6.26·45-s − 7.09·47-s − 6.66·49-s + 12.4·51-s + 4.30·53-s + 2.57·55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 0.447·5-s + 0.217·7-s + 2.08·9-s − 0.776·11-s − 0.129·13-s + 0.785·15-s − 0.991·17-s − 0.229·19-s − 0.381·21-s − 0.315·23-s + 0.200·25-s − 1.90·27-s − 0.758·29-s + 1.78·31-s + 1.36·33-s − 0.0971·35-s − 1.36·37-s + 0.228·39-s − 0.286·41-s + 0.0876·43-s − 0.933·45-s − 1.03·47-s − 0.952·49-s + 1.74·51-s + 0.591·53-s + 0.347·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5494357031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5494357031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 - 0.574T + 7T^{2} \) |
| 11 | \( 1 + 2.57T + 11T^{2} \) |
| 13 | \( 1 + 0.468T + 13T^{2} \) |
| 17 | \( 1 + 4.08T + 17T^{2} \) |
| 23 | \( 1 + 1.51T + 23T^{2} \) |
| 29 | \( 1 + 4.08T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 + 8.30T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 - 0.574T + 43T^{2} \) |
| 47 | \( 1 + 7.09T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.796456068103501627805665896735, −8.557779609975393004841146173621, −7.73732360280879317746975420588, −6.78640637620653242973433832133, −6.28135741490770889778501368230, −5.15871255454274716652356007146, −4.81012895611876814905395562315, −3.72529974220118549667073247480, −2.09957959018038073729488265120, −0.55766822594130104201468725597,
0.55766822594130104201468725597, 2.09957959018038073729488265120, 3.72529974220118549667073247480, 4.81012895611876814905395562315, 5.15871255454274716652356007146, 6.28135741490770889778501368230, 6.78640637620653242973433832133, 7.73732360280879317746975420588, 8.557779609975393004841146173621, 9.796456068103501627805665896735
