| L(s) = 1 | + 29.8·3-s − 21·5-s + 645.·9-s − 331.·11-s + 66.8·13-s − 625.·15-s + 240.·17-s − 441.·19-s + 1.07e3·23-s − 2.68e3·25-s + 1.19e4·27-s + 1.79e3·29-s + 5.68e3·31-s − 9.89e3·33-s + 1.12e4·37-s + 1.99e3·39-s + 1.20e4·41-s + 9.92e3·43-s − 1.35e4·45-s + 1.68e4·47-s + 7.16e3·51-s + 5.29e3·53-s + 6.96e3·55-s − 1.31e4·57-s − 4.13e4·59-s − 2.15e4·61-s − 1.40e3·65-s + ⋯ |
| L(s) = 1 | + 1.91·3-s − 0.375·5-s + 2.65·9-s − 0.826·11-s + 0.109·13-s − 0.718·15-s + 0.201·17-s − 0.280·19-s + 0.422·23-s − 0.858·25-s + 3.16·27-s + 0.395·29-s + 1.06·31-s − 1.58·33-s + 1.34·37-s + 0.209·39-s + 1.12·41-s + 0.818·43-s − 0.997·45-s + 1.11·47-s + 0.385·51-s + 0.259·53-s + 0.310·55-s − 0.537·57-s − 1.54·59-s − 0.740·61-s − 0.0411·65-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
| \(L(3)\) |
\(\approx\) |
\(4.724690413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.724690413\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 29.8T + 243T^{2} \) |
| 5 | \( 1 + 21T + 3.12e3T^{2} \) |
| 11 | \( 1 + 331.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 66.8T + 3.71e5T^{2} \) |
| 17 | \( 1 - 240.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 441.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.07e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.12e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.92e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.29e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.99e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.39e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.24e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.40e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.18e4T + 8.58e9T^{2} \) |
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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
−9.386328555661137466794587455250, −8.629000350234050557449025953754, −7.80542821429889573465511720678, −7.49475661433990486094304128338, −6.19480858536123440781376662154, −4.68266917979253238955802365286, −3.90839344006863622562427890211, −2.90364524360692666053296665477, −2.25744959001926327203080281883, −0.927685826378288273855472556890,
0.927685826378288273855472556890, 2.25744959001926327203080281883, 2.90364524360692666053296665477, 3.90839344006863622562427890211, 4.68266917979253238955802365286, 6.19480858536123440781376662154, 7.49475661433990486094304128338, 7.80542821429889573465511720678, 8.629000350234050557449025953754, 9.386328555661137466794587455250
