| L(s) = 1 | + 19.7·3-s − 25·5-s − 49·7-s + 147.·9-s + 292.·11-s − 1.04e3·13-s − 493.·15-s + 554.·17-s − 460.·19-s − 967.·21-s + 1.42e3·23-s + 625·25-s − 1.89e3·27-s − 7.92e3·29-s − 1.96e3·31-s + 5.78e3·33-s + 1.22e3·35-s − 1.22e4·37-s − 2.07e4·39-s − 1.35e4·41-s − 8.69e3·43-s − 3.67e3·45-s + 2.60e4·47-s + 2.40e3·49-s + 1.09e4·51-s − 4.43e3·53-s − 7.32e3·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.447·5-s − 0.377·7-s + 0.605·9-s + 0.729·11-s − 1.72·13-s − 0.566·15-s + 0.464·17-s − 0.292·19-s − 0.478·21-s + 0.563·23-s + 0.200·25-s − 0.499·27-s − 1.75·29-s − 0.367·31-s + 0.924·33-s + 0.169·35-s − 1.47·37-s − 2.18·39-s − 1.25·41-s − 0.716·43-s − 0.270·45-s + 1.72·47-s + 0.142·49-s + 0.589·51-s − 0.216·53-s − 0.326·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 25T \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 19.7T + 243T^{2} \) |
| 11 | \( 1 - 292.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.04e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 554.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 460.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.92e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.96e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.43e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 463.T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.98e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.12e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.32e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−10.28972728847681701052062724248, −9.381468707346887029422836642450, −8.737191306083918888425309500029, −7.59879261682903142408505148942, −6.97680628526731628216939458375, −5.33208753085653283680864926882, −3.96162574036797707854998730945, −3.06970809611890698964527742662, −1.90127606825536623180477890571, 0,
1.90127606825536623180477890571, 3.06970809611890698964527742662, 3.96162574036797707854998730945, 5.33208753085653283680864926882, 6.97680628526731628216939458375, 7.59879261682903142408505148942, 8.737191306083918888425309500029, 9.381468707346887029422836642450, 10.28972728847681701052062724248
