| L(s) = 1 | + 14.9·3-s − 34.9·5-s − 19.1·9-s − 748.·11-s − 12.4·13-s − 523.·15-s + 2.21e3·17-s + 1.35e3·19-s − 3.02e3·23-s − 1.90e3·25-s − 3.92e3·27-s + 5.02e3·29-s − 4.55e3·31-s − 1.11e4·33-s − 1.04e3·37-s − 186.·39-s + 3.24e3·41-s + 2.86e3·43-s + 669.·45-s + 2.05e4·47-s + 3.31e4·51-s − 2.20e4·53-s + 2.61e4·55-s + 2.03e4·57-s − 1.88e4·59-s + 5.62e4·61-s + 436.·65-s + ⋯ |
| L(s) = 1 | + 0.959·3-s − 0.625·5-s − 0.0786·9-s − 1.86·11-s − 0.0204·13-s − 0.600·15-s + 1.85·17-s + 0.862·19-s − 1.19·23-s − 0.608·25-s − 1.03·27-s + 1.10·29-s − 0.851·31-s − 1.78·33-s − 0.125·37-s − 0.0196·39-s + 0.301·41-s + 0.236·43-s + 0.0492·45-s + 1.35·47-s + 1.78·51-s − 1.07·53-s + 1.16·55-s + 0.827·57-s − 0.705·59-s + 1.93·61-s + 0.0128·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.018014720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018014720\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 14.9T + 243T^{2} \) |
| 5 | \( 1 + 34.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 748.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 12.4T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.35e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.24e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.86e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.05e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.44e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.72e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.87e4T + 8.58e9T^{2} \) |
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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
−9.612177553404070233440197304701, −8.407315953104697870785575093956, −7.82324353666045478060152913350, −7.50641602870707467503818495041, −5.84008824497677940850339044713, −5.14582210140878800703956235036, −3.76957749205374689146933615675, −3.06880480942162575780622449277, −2.16778901206258942046495967187, −0.59598900651526776494752887366,
0.59598900651526776494752887366, 2.16778901206258942046495967187, 3.06880480942162575780622449277, 3.76957749205374689146933615675, 5.14582210140878800703956235036, 5.84008824497677940850339044713, 7.50641602870707467503818495041, 7.82324353666045478060152913350, 8.407315953104697870785575093956, 9.612177553404070233440197304701
