| L(s) = 1 | − 3.23·3-s − 312.·5-s + 766.·7-s − 2.17e3·9-s − 252.·11-s − 1.06e3·13-s + 1.01e3·15-s − 1.87e4·17-s + 6.85e3·19-s − 2.48e3·21-s − 1.15e4·23-s + 1.95e4·25-s + 1.41e4·27-s − 4.62e4·29-s − 4.68e4·31-s + 816.·33-s − 2.39e5·35-s − 1.82e5·37-s + 3.45e3·39-s + 8.19e5·41-s − 4.77e5·43-s + 6.80e5·45-s − 9.92e5·47-s − 2.35e5·49-s + 6.07e4·51-s − 8.52e5·53-s + 7.87e4·55-s + ⋯ |
| L(s) = 1 | − 0.0692·3-s − 1.11·5-s + 0.844·7-s − 0.995·9-s − 0.0570·11-s − 0.134·13-s + 0.0774·15-s − 0.926·17-s + 0.229·19-s − 0.0585·21-s − 0.198·23-s + 0.249·25-s + 0.138·27-s − 0.352·29-s − 0.282·31-s + 0.00395·33-s − 0.944·35-s − 0.593·37-s + 0.00931·39-s + 1.85·41-s − 0.915·43-s + 1.11·45-s − 1.39·47-s − 0.286·49-s + 0.0641·51-s − 0.786·53-s + 0.0638·55-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
| \(L(4)\) |
\(\approx\) |
\(1.030444912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030444912\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 3.23T + 2.18e3T^{2} \) |
| 5 | \( 1 + 312.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 766.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 252.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.06e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.87e4T + 4.10e8T^{2} \) |
| 23 | \( 1 + 1.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.62e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.68e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.82e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.77e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.92e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 8.52e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.92e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.09e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.32e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.37e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.30e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 3.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.90e6T + 8.07e13T^{2} \) |
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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
−10.90298893599899128511606111103, −9.435665292219585507758942729541, −8.361544269113017514538357046938, −7.86671574026825351601757398200, −6.71163363018131659653850396240, −5.41740733050567321746290065540, −4.45437369912080224557025067010, −3.37164009701460617257009795158, −2.05075396818652840545846865378, −0.48038851734849477595794653620,
0.48038851734849477595794653620, 2.05075396818652840545846865378, 3.37164009701460617257009795158, 4.45437369912080224557025067010, 5.41740733050567321746290065540, 6.71163363018131659653850396240, 7.86671574026825351601757398200, 8.361544269113017514538357046938, 9.435665292219585507758942729541, 10.90298893599899128511606111103
