L(s) = 1 | + 6.89·3-s + 48.4·5-s − 195.·9-s + 163.·11-s + 120.·13-s + 334.·15-s − 78.1·17-s − 2.26e3·19-s + 2.45e3·23-s − 776.·25-s − 3.02e3·27-s + 6.98e3·29-s + 2.79e3·31-s + 1.12e3·33-s + 9.45e3·37-s + 832.·39-s − 1.00e4·41-s + 6.93e3·43-s − 9.47e3·45-s + 1.16e3·47-s − 538.·51-s + 8.56e3·53-s + 7.92e3·55-s − 1.56e4·57-s + 6.22e3·59-s + 4.19e4·61-s + 5.85e3·65-s + ⋯ |
L(s) = 1 | + 0.442·3-s + 0.866·5-s − 0.804·9-s + 0.407·11-s + 0.198·13-s + 0.383·15-s − 0.0656·17-s − 1.43·19-s + 0.966·23-s − 0.248·25-s − 0.797·27-s + 1.54·29-s + 0.522·31-s + 0.180·33-s + 1.13·37-s + 0.0876·39-s − 0.937·41-s + 0.571·43-s − 0.697·45-s + 0.0769·47-s − 0.0290·51-s + 0.418·53-s + 0.353·55-s − 0.636·57-s + 0.232·59-s + 1.44·61-s + 0.171·65-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
\(L(3)\) |
\(\approx\) |
\(2.990362635\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.990362635\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 6.89T + 243T^{2} \) |
| 5 | \( 1 - 48.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 120.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 78.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.00e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.16e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 6.22e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.81e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.49e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.45e5T + 8.58e9T^{2} \) |
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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
−9.451829956363657024653924548405, −8.693338469435825687156315515797, −8.099598967670141187943732271770, −6.72950774177481179857760282477, −6.14474069399899338652475443250, −5.14689426228218125915424948791, −4.03419559650972499618254299015, −2.83221284806293859105693043651, −2.07029295629872772380746580973, −0.77119680208208657256980920474,
0.77119680208208657256980920474, 2.07029295629872772380746580973, 2.83221284806293859105693043651, 4.03419559650972499618254299015, 5.14689426228218125915424948791, 6.14474069399899338652475443250, 6.72950774177481179857760282477, 8.099598967670141187943732271770, 8.693338469435825687156315515797, 9.451829956363657024653924548405