L(s) = 1 | + 16·2-s − 123.·3-s + 256·4-s + 1.27e3·5-s − 1.97e3·6-s + 4.09e3·8-s − 4.41e3·9-s + 2.04e4·10-s − 2.76e4·11-s − 3.16e4·12-s − 8.33e4·13-s − 1.57e5·15-s + 6.55e4·16-s + 5.54e5·17-s − 7.07e4·18-s − 3.43e5·19-s + 3.26e5·20-s − 4.42e5·22-s − 5.35e5·23-s − 5.06e5·24-s − 3.23e5·25-s − 1.33e6·26-s + 2.97e6·27-s − 2.59e6·29-s − 2.52e6·30-s + 6.09e6·31-s + 1.04e6·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.880·3-s + 0.5·4-s + 0.913·5-s − 0.622·6-s + 0.353·8-s − 0.224·9-s + 0.645·10-s − 0.569·11-s − 0.440·12-s − 0.809·13-s − 0.804·15-s + 0.250·16-s + 1.60·17-s − 0.158·18-s − 0.604·19-s + 0.456·20-s − 0.402·22-s − 0.399·23-s − 0.311·24-s − 0.165·25-s − 0.572·26-s + 1.07·27-s − 0.680·29-s − 0.568·30-s + 1.18·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 16T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 123.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.27e3T + 1.95e6T^{2} \) |
| 11 | \( 1 + 2.76e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 8.33e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.54e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.43e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.35e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.59e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.09e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.78e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.05e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 8.75e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.72e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 8.34e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.15e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.98e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.85e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.70e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.46e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.19e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.98e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.80e8T + 7.60e17T^{2} \) |
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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
−11.82236854078292406429467246080, −10.55728209305113004763534349412, −9.815060782048062711010606091129, −8.048047927829842809235348544844, −6.60159460989186824240392611696, −5.62952361146217294659237088467, −4.94922883066078126700334948675, −3.11480563363481081390916390525, −1.72991262628657713484677009694, 0,
1.72991262628657713484677009694, 3.11480563363481081390916390525, 4.94922883066078126700334948675, 5.62952361146217294659237088467, 6.60159460989186824240392611696, 8.048047927829842809235348544844, 9.815060782048062711010606091129, 10.55728209305113004763534349412, 11.82236854078292406429467246080