| L(s) = 1 | + 13.4·3-s + 25·5-s − 138.·7-s − 62.9·9-s + 259.·11-s − 154·13-s + 335.·15-s + 178·17-s + 965.·19-s − 1.86e3·21-s − 2.63e3·23-s + 625·25-s − 4.10e3·27-s − 4.11e3·29-s − 3.15e3·31-s + 3.48e3·33-s − 3.46e3·35-s − 7.44e3·37-s − 2.06e3·39-s + 7.27e3·41-s − 1.79e4·43-s − 1.57e3·45-s + 7.41e3·47-s + 2.41e3·49-s + 2.38e3·51-s − 3.22e4·53-s + 6.48e3·55-s + ⋯ |
| L(s) = 1 | + 0.860·3-s + 0.447·5-s − 1.06·7-s − 0.259·9-s + 0.646·11-s − 0.252·13-s + 0.384·15-s + 0.149·17-s + 0.613·19-s − 0.920·21-s − 1.03·23-s + 0.200·25-s − 1.08·27-s − 0.907·29-s − 0.590·31-s + 0.556·33-s − 0.478·35-s − 0.893·37-s − 0.217·39-s + 0.675·41-s − 1.47·43-s − 0.115·45-s + 0.489·47-s + 0.143·49-s + 0.128·51-s − 1.57·53-s + 0.289·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 - 13.4T + 243T^{2} \) |
| 7 | \( 1 + 138.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 259.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 154T + 3.71e5T^{2} \) |
| 17 | \( 1 - 178T + 1.41e6T^{2} \) |
| 19 | \( 1 - 965.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.44e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.27e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.79e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.41e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.98e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.08e5T + 8.58e9T^{2} \) |
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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
−9.945554062805176147158495009041, −9.458577640981746493848796912294, −8.600973770804502783041058379535, −7.49448363311508899974509643248, −6.43465355029748116204503022177, −5.47640448027724189083629960911, −3.82779672076084275475475817841, −3.00121562687584385147275153389, −1.78668976466563719064047722148, 0,
1.78668976466563719064047722148, 3.00121562687584385147275153389, 3.82779672076084275475475817841, 5.47640448027724189083629960911, 6.43465355029748116204503022177, 7.49448363311508899974509643248, 8.600973770804502783041058379535, 9.458577640981746493848796912294, 9.945554062805176147158495009041
