| L(s) = 1 | − 18.5·3-s − 57.9·5-s + 101.·9-s − 271.·11-s + 25.9·13-s + 1.07e3·15-s + 65.8·17-s + 1.60e3·19-s − 3.82e3·23-s + 229.·25-s + 2.62e3·27-s − 4.45e3·29-s − 8.91e3·31-s + 5.04e3·33-s − 1.49e4·37-s − 482.·39-s − 1.27e4·41-s + 3.06e3·43-s − 5.89e3·45-s + 1.61e4·47-s − 1.22e3·51-s − 3.15e4·53-s + 1.57e4·55-s − 2.97e4·57-s + 1.07e4·59-s − 1.26e4·61-s − 1.50e3·65-s + ⋯ |
| L(s) = 1 | − 1.19·3-s − 1.03·5-s + 0.418·9-s − 0.676·11-s + 0.0426·13-s + 1.23·15-s + 0.0552·17-s + 1.01·19-s − 1.50·23-s + 0.0734·25-s + 0.692·27-s − 0.984·29-s − 1.66·31-s + 0.805·33-s − 1.79·37-s − 0.0507·39-s − 1.18·41-s + 0.253·43-s − 0.434·45-s + 1.06·47-s − 0.0658·51-s − 1.54·53-s + 0.700·55-s − 1.21·57-s + 0.403·59-s − 0.433·61-s − 0.0441·65-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
| \(L(3)\) |
\(\approx\) |
\(0.04064307401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04064307401\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 18.5T + 243T^{2} \) |
| 5 | \( 1 + 57.9T + 3.12e3T^{2} \) |
| 11 | \( 1 + 271.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 25.9T + 3.71e5T^{2} \) |
| 17 | \( 1 - 65.8T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.82e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.45e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.06e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.73e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.92e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.46e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.92e4T + 8.58e9T^{2} \) |
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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
−9.714735797143394970814932363993, −8.561805986662705566893498655158, −7.66216037632087331031625157079, −7.03020215278208196724950778874, −5.78584629284553057217610500715, −5.29560546209674395003628606319, −4.16538724092979968138585134165, −3.24537443028563764988451819004, −1.65096467982233172634123681312, −0.097176759799663324425566179620,
0.097176759799663324425566179620, 1.65096467982233172634123681312, 3.24537443028563764988451819004, 4.16538724092979968138585134165, 5.29560546209674395003628606319, 5.78584629284553057217610500715, 7.03020215278208196724950778874, 7.66216037632087331031625157079, 8.561805986662705566893498655158, 9.714735797143394970814932363993
