| L(s) = 1 | + 8.09·3-s + 276.·5-s − 720.·7-s − 2.12e3·9-s − 4.97e3·11-s + 8.08e3·13-s + 2.23e3·15-s + 1.37e3·17-s − 6.85e3·19-s − 5.83e3·21-s − 5.50e4·23-s − 1.76e3·25-s − 3.48e4·27-s − 1.17e5·29-s − 1.51e5·31-s − 4.02e4·33-s − 1.99e5·35-s − 7.60e4·37-s + 6.54e4·39-s + 4.20e5·41-s − 4.11e5·43-s − 5.86e5·45-s − 7.91e5·47-s − 3.03e5·49-s + 1.11e4·51-s − 8.00e5·53-s − 1.37e6·55-s + ⋯ |
| L(s) = 1 | + 0.173·3-s + 0.988·5-s − 0.794·7-s − 0.970·9-s − 1.12·11-s + 1.02·13-s + 0.171·15-s + 0.0679·17-s − 0.229·19-s − 0.137·21-s − 0.943·23-s − 0.0225·25-s − 0.340·27-s − 0.896·29-s − 0.910·31-s − 0.194·33-s − 0.785·35-s − 0.246·37-s + 0.176·39-s + 0.953·41-s − 0.789·43-s − 0.959·45-s − 1.11·47-s − 0.369·49-s + 0.0117·51-s − 0.738·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 6.85e3T \) |
| good | 3 | \( 1 - 8.09T + 2.18e3T^{2} \) |
| 5 | \( 1 - 276.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 720.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.97e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.08e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.37e3T + 4.10e8T^{2} \) |
| 23 | \( 1 + 5.50e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.17e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.51e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 7.60e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.11e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.00e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 3.09e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.19e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.20e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.69e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.98e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.62e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.86e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.21e7T + 8.07e13T^{2} \) |
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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
−12.86218786378153143206425250425, −11.28606520353128882358194508217, −10.19029757647248984516559447459, −9.183490035641235671428248284628, −7.998995673142444779296141374813, −6.28562746600711123647841210183, −5.48429504003763877122652506159, −3.40615805873004771621698979568, −2.08106368941208712746800657076, 0,
2.08106368941208712746800657076, 3.40615805873004771621698979568, 5.48429504003763877122652506159, 6.28562746600711123647841210183, 7.998995673142444779296141374813, 9.183490035641235671428248284628, 10.19029757647248984516559447459, 11.28606520353128882358194508217, 12.86218786378153143206425250425
