| L(s) = 1 | + 84.8·3-s + 72.4·5-s + 210.·7-s + 5.02e3·9-s − 2.35e3·11-s + 2.21e3·13-s + 6.15e3·15-s + 2.24e4·17-s + 6.85e3·19-s + 1.78e4·21-s + 3.38e4·23-s − 7.28e4·25-s + 2.40e5·27-s − 9.36e4·29-s + 2.22e5·31-s − 1.99e5·33-s + 1.52e4·35-s + 2.28e5·37-s + 1.88e5·39-s + 6.73e4·41-s − 1.62e5·43-s + 3.63e5·45-s − 8.16e4·47-s − 7.79e5·49-s + 1.90e6·51-s + 9.05e5·53-s − 1.70e5·55-s + ⋯ |
| L(s) = 1 | + 1.81·3-s + 0.259·5-s + 0.231·7-s + 2.29·9-s − 0.532·11-s + 0.280·13-s + 0.470·15-s + 1.10·17-s + 0.229·19-s + 0.420·21-s + 0.579·23-s − 0.932·25-s + 2.35·27-s − 0.713·29-s + 1.34·31-s − 0.966·33-s + 0.0600·35-s + 0.742·37-s + 0.508·39-s + 0.152·41-s − 0.312·43-s + 0.595·45-s − 0.114·47-s − 0.946·49-s + 2.00·51-s + 0.835·53-s − 0.138·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.355259217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.355259217\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 84.8T + 2.18e3T^{2} \) |
| 5 | \( 1 - 72.4T + 7.81e4T^{2} \) |
| 7 | \( 1 - 210.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.35e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 2.21e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.24e4T + 4.10e8T^{2} \) |
| 23 | \( 1 - 3.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.36e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.28e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.73e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.62e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.16e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.05e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.18e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.73e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.22e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.34e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.68e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 6.74e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.83e5T + 8.07e13T^{2} \) |
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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
−10.01591304469538321475784495468, −9.620621529796598049917111115057, −8.422153014577789342698125640994, −7.951653427812015973512400926402, −6.96500029265525897436603018420, −5.45730669705339249633925586178, −4.11499797330727975125547068589, −3.14007306604022485109688800318, −2.23995864856812135571020778222, −1.12007105681502797905691612403,
1.12007105681502797905691612403, 2.23995864856812135571020778222, 3.14007306604022485109688800318, 4.11499797330727975125547068589, 5.45730669705339249633925586178, 6.96500029265525897436603018420, 7.951653427812015973512400926402, 8.422153014577789342698125640994, 9.620621529796598049917111115057, 10.01591304469538321475784495468
