| L(s) = 1 | − 14.5·3-s + 44.8·5-s − 31.8·9-s + 634.·11-s + 20.5·13-s − 651.·15-s + 1.05e3·17-s + 65.1·19-s + 4.05e3·23-s − 1.11e3·25-s + 3.99e3·27-s − 6.58e3·29-s + 1.08e3·31-s − 9.22e3·33-s + 685.·37-s − 299.·39-s + 1.36e4·41-s + 1.60e4·43-s − 1.42e3·45-s − 1.36e4·47-s − 1.52e4·51-s − 1.60e3·53-s + 2.84e4·55-s − 947.·57-s − 7.52e3·59-s − 2.08e4·61-s + 922.·65-s + ⋯ |
| L(s) = 1 | − 0.932·3-s + 0.801·5-s − 0.131·9-s + 1.58·11-s + 0.0337·13-s − 0.747·15-s + 0.881·17-s + 0.0414·19-s + 1.59·23-s − 0.357·25-s + 1.05·27-s − 1.45·29-s + 0.203·31-s − 1.47·33-s + 0.0822·37-s − 0.0314·39-s + 1.26·41-s + 1.32·43-s − 0.105·45-s − 0.898·47-s − 0.821·51-s − 0.0786·53-s + 1.26·55-s − 0.0386·57-s − 0.281·59-s − 0.718·61-s + 0.0270·65-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
| \(L(3)\) |
\(\approx\) |
\(2.114337169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114337169\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 14.5T + 243T^{2} \) |
| 5 | \( 1 - 44.8T + 3.12e3T^{2} \) |
| 11 | \( 1 - 634.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 20.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.05e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 65.1T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 685.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.36e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.60e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.52e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.87e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.76e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.02e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.68e5T + 8.58e9T^{2} \) |
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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
−9.387717883170803795640249365574, −9.074185450060133436116798162339, −7.67284553781467461803110696209, −6.68015199049403576566212348992, −5.96342813973122815780527136575, −5.36033802716560963953049819738, −4.21773441132992560127764495585, −3.03525348994828925075713239569, −1.61291772081948979295399985623, −0.74829725557683630368287717227,
0.74829725557683630368287717227, 1.61291772081948979295399985623, 3.03525348994828925075713239569, 4.21773441132992560127764495585, 5.36033802716560963953049819738, 5.96342813973122815780527136575, 6.68015199049403576566212348992, 7.67284553781467461803110696209, 9.074185450060133436116798162339, 9.387717883170803795640249365574
