| L(s) = 1 | + 33.1·2-s − 71.4·3-s + 584.·4-s − 957.·5-s − 2.36e3·6-s − 5.22e3·7-s + 2.41e3·8-s − 1.45e4·9-s − 3.17e4·10-s + 4.12e4·11-s − 4.17e4·12-s − 7.00e4·13-s − 1.73e5·14-s + 6.84e4·15-s − 2.19e5·16-s − 5.26e5·17-s − 4.82e5·18-s + 7.90e5·19-s − 5.59e5·20-s + 3.73e5·21-s + 1.36e6·22-s + 7.03e5·23-s − 1.72e5·24-s − 1.03e6·25-s − 2.31e6·26-s + 2.44e6·27-s − 3.05e6·28-s + ⋯ |
| L(s) = 1 | + 1.46·2-s − 0.509·3-s + 1.14·4-s − 0.685·5-s − 0.745·6-s − 0.823·7-s + 0.208·8-s − 0.740·9-s − 1.00·10-s + 0.850·11-s − 0.581·12-s − 0.679·13-s − 1.20·14-s + 0.349·15-s − 0.837·16-s − 1.52·17-s − 1.08·18-s + 1.39·19-s − 0.782·20-s + 0.419·21-s + 1.24·22-s + 0.523·23-s − 0.106·24-s − 0.530·25-s − 0.995·26-s + 0.886·27-s − 0.940·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + 7.07e5T \) |
| good | 2 | \( 1 - 33.1T + 512T^{2} \) |
| 3 | \( 1 + 71.4T + 1.96e4T^{2} \) |
| 5 | \( 1 + 957.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 5.22e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 4.12e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.00e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.26e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.90e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.03e5T + 1.80e12T^{2} \) |
| 31 | \( 1 + 2.33e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 7.47e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 9.03e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.06e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.45e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.91e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.85e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.88e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.18e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.25e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.02e9T + 3.50e17T^{2} \) |
| 97 | \( 1 - 6.04e8T + 7.60e17T^{2} \) |
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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
−14.33616274289663815820244232146, −13.16043394276819693842111581838, −11.98290441710732652104241090121, −11.30965469812894037521925244842, −9.200279692235728941921770541366, −6.98958141249011514007471494403, −5.77255318906166498460090321963, −4.32274682330421634734261732387, −2.96567409446422918240914077092, 0,
2.96567409446422918240914077092, 4.32274682330421634734261732387, 5.77255318906166498460090321963, 6.98958141249011514007471494403, 9.200279692235728941921770541366, 11.30965469812894037521925244842, 11.98290441710732652104241090121, 13.16043394276819693842111581838, 14.33616274289663815820244232146
