| L(s) = 1 | − 27·3-s + 514.·5-s + 1.60e3·7-s + 729·9-s − 6.91e3·11-s + 1.23e3·13-s − 1.38e4·15-s + 2.71e4·17-s + 4.41e4·19-s − 4.32e4·21-s − 1.55e4·23-s + 1.86e5·25-s − 1.96e4·27-s − 9.45e4·29-s + 9.79e4·31-s + 1.86e5·33-s + 8.23e5·35-s + 5.56e5·37-s − 3.34e4·39-s − 2.03e5·41-s − 5.95e4·43-s + 3.74e5·45-s − 6.98e5·47-s + 1.73e6·49-s − 7.32e5·51-s − 9.16e4·53-s − 3.55e6·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.84·5-s + 1.76·7-s + 0.333·9-s − 1.56·11-s + 0.156·13-s − 1.06·15-s + 1.33·17-s + 1.47·19-s − 1.01·21-s − 0.265·23-s + 2.38·25-s − 0.192·27-s − 0.720·29-s + 0.590·31-s + 0.904·33-s + 3.24·35-s + 1.80·37-s − 0.0902·39-s − 0.460·41-s − 0.114·43-s + 0.613·45-s − 0.981·47-s + 2.11·49-s − 0.772·51-s − 0.0845·53-s − 2.88·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.687844661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.687844661\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 27T \) |
| good | 5 | \( 1 - 514.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.60e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.91e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.23e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.71e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.41e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.55e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.45e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 9.79e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.56e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.95e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 6.98e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.16e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.47e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 4.24e4T + 3.14e12T^{2} \) |
| 67 | \( 1 + 7.22e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.41e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.12e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.42e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.55e6T + 8.07e13T^{2} \) |
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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
−10.17336392269234139810622108345, −9.535452896378048749666869226196, −8.151459508458390675003002948264, −7.48117929704084284921628876282, −5.95369404784908382296830774469, −5.36712953585746690102311733656, −4.84059617212196700263392833672, −2.82648197022283686964580929431, −1.74962375568427336776693155228, −1.01271362503436043365370149704,
1.01271362503436043365370149704, 1.74962375568427336776693155228, 2.82648197022283686964580929431, 4.84059617212196700263392833672, 5.36712953585746690102311733656, 5.95369404784908382296830774469, 7.48117929704084284921628876282, 8.151459508458390675003002948264, 9.535452896378048749666869226196, 10.17336392269234139810622108345
