| L(s) = 1 | − 47.8·3-s − 109.·5-s − 5.94e3·7-s − 1.73e4·9-s + 2.52e4·11-s + 2.85e4·13-s + 5.25e3·15-s + 1.09e5·17-s + 9.04e5·19-s + 2.84e5·21-s + 4.35e5·23-s − 1.94e6·25-s + 1.77e6·27-s + 6.44e6·29-s − 6.62e6·31-s − 1.20e6·33-s + 6.52e5·35-s + 4.14e6·37-s − 1.36e6·39-s + 1.49e7·41-s − 4.01e7·43-s + 1.90e6·45-s − 6.30e6·47-s − 4.98e6·49-s − 5.23e6·51-s + 1.53e7·53-s − 2.76e6·55-s + ⋯ |
| L(s) = 1 | − 0.341·3-s − 0.0785·5-s − 0.936·7-s − 0.883·9-s + 0.519·11-s + 0.277·13-s + 0.0268·15-s + 0.317·17-s + 1.59·19-s + 0.319·21-s + 0.324·23-s − 0.993·25-s + 0.642·27-s + 1.69·29-s − 1.28·31-s − 0.177·33-s + 0.0735·35-s + 0.363·37-s − 0.0946·39-s + 0.826·41-s − 1.79·43-s + 0.0693·45-s − 0.188·47-s − 0.123·49-s − 0.108·51-s + 0.266·53-s − 0.0407·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 + 47.8T + 1.96e4T^{2} \) |
| 5 | \( 1 + 109.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.94e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.52e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.44e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.62e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.14e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.49e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 4.01e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.30e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.53e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.52e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 8.66e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.01e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.14e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.00e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.34e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.47e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.25e9T + 7.60e17T^{2} \) |
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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
−10.17141166420217528870790884133, −9.365241707898519012143571410896, −8.326281808654221457938322123238, −7.08077124576491817852382779598, −6.11263202233780018088132064070, −5.23371043423504310761354920702, −3.70088849799796462963140336257, −2.82386876762059450946972992543, −1.13933177173345191038160784256, 0,
1.13933177173345191038160784256, 2.82386876762059450946972992543, 3.70088849799796462963140336257, 5.23371043423504310761354920702, 6.11263202233780018088132064070, 7.08077124576491817852382779598, 8.326281808654221457938322123238, 9.365241707898519012143571410896, 10.17141166420217528870790884133
