| L(s) = 1 | − 27·3-s + 480.·5-s + 729·9-s + 8.12e3·11-s − 1.22e4·13-s − 1.29e4·15-s + 3.16e4·17-s + 8.81e3·19-s + 1.43e4·23-s + 1.52e5·25-s − 1.96e4·27-s − 1.04e5·29-s − 5.73e4·31-s − 2.19e5·33-s + 1.28e5·37-s + 3.30e5·39-s + 5.59e5·41-s + 6.02e5·43-s + 3.50e5·45-s + 3.75e5·47-s − 8.55e5·51-s + 9.42e4·53-s + 3.90e6·55-s − 2.38e5·57-s − 4.87e4·59-s + 5.15e5·61-s − 5.88e6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.71·5-s + 0.333·9-s + 1.83·11-s − 1.54·13-s − 0.992·15-s + 1.56·17-s + 0.294·19-s + 0.245·23-s + 1.95·25-s − 0.192·27-s − 0.792·29-s − 0.346·31-s − 1.06·33-s + 0.416·37-s + 0.891·39-s + 1.26·41-s + 1.15·43-s + 0.573·45-s + 0.527·47-s − 0.903·51-s + 0.0869·53-s + 3.16·55-s − 0.170·57-s − 0.0309·59-s + 0.290·61-s − 2.65·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.408852482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.408852482\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 480.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 8.12e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.81e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.43e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.73e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.28e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 6.02e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 3.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.42e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.87e4T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.15e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.01e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.20e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.90e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.39e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 5.39e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.41e6T + 8.07e13T^{2} \) |
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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
−9.507084179712579698398966721847, −9.227956668991035900541263079643, −7.55630062472326027638665741153, −6.73760624191585831143159921758, −5.83609863126090916423827442117, −5.31157033417808555295764374160, −4.10849444633271529013245897869, −2.70733279728920757329300747016, −1.62977306497517793835695145950, −0.877350864726866173261547107895,
0.877350864726866173261547107895, 1.62977306497517793835695145950, 2.70733279728920757329300747016, 4.10849444633271529013245897869, 5.31157033417808555295764374160, 5.83609863126090916423827442117, 6.73760624191585831143159921758, 7.55630062472326027638665741153, 9.227956668991035900541263079643, 9.507084179712579698398966721847
