| L(s) = 1 | − 10.0·3-s + 79.8·5-s − 141.·9-s − 351.·11-s + 291.·13-s − 804.·15-s + 370.·17-s + 1.50e3·19-s + 425.·23-s + 3.24e3·25-s + 3.87e3·27-s − 7.78e3·29-s − 2.57e3·31-s + 3.54e3·33-s + 739.·37-s − 2.94e3·39-s − 7.02e3·41-s − 1.83e3·43-s − 1.12e4·45-s − 1.53e3·47-s − 3.73e3·51-s − 9.53e3·53-s − 2.80e4·55-s − 1.51e4·57-s − 2.96e4·59-s + 4.65e4·61-s + 2.32e4·65-s + ⋯ |
| L(s) = 1 | − 0.646·3-s + 1.42·5-s − 0.581·9-s − 0.876·11-s + 0.478·13-s − 0.923·15-s + 0.310·17-s + 0.956·19-s + 0.167·23-s + 1.03·25-s + 1.02·27-s − 1.71·29-s − 0.481·31-s + 0.567·33-s + 0.0888·37-s − 0.309·39-s − 0.653·41-s − 0.151·43-s − 0.830·45-s − 0.101·47-s − 0.200·51-s − 0.466·53-s − 1.25·55-s − 0.618·57-s − 1.10·59-s + 1.60·61-s + 0.683·65-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 10.0T + 243T^{2} \) |
| 5 | \( 1 - 79.8T + 3.12e3T^{2} \) |
| 11 | \( 1 + 351.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 291.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 370.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 425.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.78e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 739.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.83e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.53e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.65e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.00e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.03e5T + 8.58e9T^{2} \) |
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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
−9.319374064056978782981614558017, −8.329442419613786715980111085359, −7.26567632715431763854613801694, −6.19926250155117343729838837889, −5.54981920002090081228782754013, −5.08804531634692517039636807600, −3.42401624315020666490330723573, −2.36466509535289278701470419559, −1.30010534209090350147329646392, 0,
1.30010534209090350147329646392, 2.36466509535289278701470419559, 3.42401624315020666490330723573, 5.08804531634692517039636807600, 5.54981920002090081228782754013, 6.19926250155117343729838837889, 7.26567632715431763854613801694, 8.329442419613786715980111085359, 9.319374064056978782981614558017
