| L(s) = 1 | + 24.4·3-s − 83.6·5-s + 354.·9-s − 549.·11-s + 823.·13-s − 2.04e3·15-s − 145.·17-s + 1.76e3·19-s + 1.52e3·23-s + 3.87e3·25-s + 2.72e3·27-s − 741.·29-s − 3.20e3·31-s − 1.34e4·33-s − 3.54e3·37-s + 2.01e4·39-s + 6.46e3·41-s − 6.71e3·43-s − 2.96e4·45-s − 1.91e4·47-s − 3.55e3·51-s − 2.12e4·53-s + 4.59e4·55-s + 4.32e4·57-s + 3.65e4·59-s − 4.34e4·61-s − 6.88e4·65-s + ⋯ |
| L(s) = 1 | + 1.56·3-s − 1.49·5-s + 1.45·9-s − 1.36·11-s + 1.35·13-s − 2.34·15-s − 0.122·17-s + 1.12·19-s + 0.600·23-s + 1.23·25-s + 0.718·27-s − 0.163·29-s − 0.598·31-s − 2.14·33-s − 0.426·37-s + 2.11·39-s + 0.600·41-s − 0.553·43-s − 2.18·45-s − 1.26·47-s − 0.191·51-s − 1.04·53-s + 2.04·55-s + 1.76·57-s + 1.36·59-s − 1.49·61-s − 2.02·65-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 | \( 1 - 24.4T + 243T^{2} \) |
| 5 | \( 1 + 83.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 549.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 823.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 145.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.76e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 741.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.20e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.54e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.46e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.71e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.65e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.34e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.02e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.31e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.67e4T + 8.58e9T^{2} \) |
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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
−8.836011721636142305957217887534, −8.162913273494487508703673340255, −7.75244941698651743035443226954, −6.94052033843498701832193508870, −5.37234396090611504748137970981, −4.21114134310529893649991186834, −3.38919731333108019090824196901, −2.85774593339162307987703426699, −1.41154210407477838249747041414, 0,
1.41154210407477838249747041414, 2.85774593339162307987703426699, 3.38919731333108019090824196901, 4.21114134310529893649991186834, 5.37234396090611504748137970981, 6.94052033843498701832193508870, 7.75244941698651743035443226954, 8.162913273494487508703673340255, 8.836011721636142305957217887534
