| L(s) = 1 | + 16·2-s + 256·4-s + 2.28e3·5-s − 1.36e3·7-s + 4.09e3·8-s + 3.65e4·10-s + 2.77e4·11-s + 3.80e4·13-s − 2.18e4·14-s + 6.55e4·16-s + 1.36e5·17-s + 6.04e5·19-s + 5.84e5·20-s + 4.44e5·22-s − 2.42e6·23-s + 3.26e6·25-s + 6.09e5·26-s − 3.50e5·28-s + 4.55e6·29-s − 1.10e6·31-s + 1.04e6·32-s + 2.17e6·34-s − 3.12e6·35-s + 1.63e7·37-s + 9.66e6·38-s + 9.35e6·40-s + 2.04e7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.63·5-s − 0.215·7-s + 0.353·8-s + 1.15·10-s + 0.571·11-s + 0.369·13-s − 0.152·14-s + 0.250·16-s + 0.395·17-s + 1.06·19-s + 0.817·20-s + 0.404·22-s − 1.80·23-s + 1.67·25-s + 0.261·26-s − 0.107·28-s + 1.19·29-s − 0.214·31-s + 0.176·32-s + 0.279·34-s − 0.352·35-s + 1.43·37-s + 0.752·38-s + 0.577·40-s + 1.13·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.370096942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.370096942\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 16T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.28e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.36e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.80e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.36e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 6.04e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.42e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.55e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.10e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.63e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.04e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.09e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.23e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.17e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 5.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.75e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.68e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.13e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.79e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.74e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.85e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.50e8T + 7.60e17T^{2} \) |
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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
−11.26495002600193450787902160455, −9.990579629132767059930678007892, −9.507829084513516679066136382612, −7.992998276318946691308478730639, −6.45380177653381330310568933766, −5.96890414151854248483240449063, −4.82922382840879679151919278604, −3.38139986740544299486546225692, −2.17174332016560259896739315092, −1.14692596472200272248746053565,
1.14692596472200272248746053565, 2.17174332016560259896739315092, 3.38139986740544299486546225692, 4.82922382840879679151919278604, 5.96890414151854248483240449063, 6.45380177653381330310568933766, 7.992998276318946691308478730639, 9.507829084513516679066136382612, 9.990579629132767059930678007892, 11.26495002600193450787902160455
