| L(s) = 1 | + 256·2-s − 1.49e4·3-s + 6.55e4·4-s + 3.90e5·5-s − 3.83e6·6-s + 1.48e7·7-s + 1.67e7·8-s + 9.51e7·9-s + 1.00e8·10-s − 1.08e9·11-s − 9.81e8·12-s − 4.59e9·13-s + 3.79e9·14-s − 5.85e9·15-s + 4.29e9·16-s − 1.61e10·17-s + 2.43e10·18-s + 4.80e10·19-s + 2.56e10·20-s − 2.21e11·21-s − 2.77e11·22-s − 5.71e11·23-s − 2.51e11·24-s + 1.52e11·25-s − 1.17e12·26-s + 5.09e11·27-s + 9.70e11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.31·3-s + 1/2·4-s + 0.447·5-s − 0.931·6-s + 0.970·7-s + 0.353·8-s + 0.736·9-s + 0.316·10-s − 1.52·11-s − 0.658·12-s − 1.56·13-s + 0.686·14-s − 0.589·15-s + 1/4·16-s − 0.559·17-s + 0.520·18-s + 0.649·19-s + 0.223·20-s − 1.27·21-s − 1.07·22-s − 1.52·23-s − 0.465·24-s + 1/5·25-s − 1.10·26-s + 0.346·27-s + 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{8} T \) |
| 5 | \( 1 - p^{8} T \) |
| good | 3 | \( 1 + 1664 p^{2} T + p^{17} T^{2} \) |
| 7 | \( 1 - 2115524 p T + p^{17} T^{2} \) |
| 11 | \( 1 + 8967228 p^{2} T + p^{17} T^{2} \) |
| 13 | \( 1 + 4595303746 T + p^{17} T^{2} \) |
| 17 | \( 1 + 16104698622 T + p^{17} T^{2} \) |
| 19 | \( 1 - 48093117860 T + p^{17} T^{2} \) |
| 23 | \( 1 + 571023069276 T + p^{17} T^{2} \) |
| 29 | \( 1 + 1726424788290 T + p^{17} T^{2} \) |
| 31 | \( 1 + 5623721940808 T + p^{17} T^{2} \) |
| 37 | \( 1 + 10013128639162 T + p^{17} T^{2} \) |
| 41 | \( 1 + 37505113176198 T + p^{17} T^{2} \) |
| 43 | \( 1 - 136226190448184 T + p^{17} T^{2} \) |
| 47 | \( 1 + 37681319902812 T + p^{17} T^{2} \) |
| 53 | \( 1 + 22543738268346 T + p^{17} T^{2} \) |
| 59 | \( 1 - 221363585667420 T + p^{17} T^{2} \) |
| 61 | \( 1 + 276238009706818 T + p^{17} T^{2} \) |
| 67 | \( 1 - 6165400365120968 T + p^{17} T^{2} \) |
| 71 | \( 1 + 129445634389248 T + p^{17} T^{2} \) |
| 73 | \( 1 + 9751215737952646 T + p^{17} T^{2} \) |
| 79 | \( 1 + 25538259579681280 T + p^{17} T^{2} \) |
| 83 | \( 1 - 29930106765986544 T + p^{17} T^{2} \) |
| 89 | \( 1 + 24258364335352710 T + p^{17} T^{2} \) |
| 97 | \( 1 - 123350176379809778 T + p^{17} T^{2} \) |
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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
−15.91083813137086685485187029150, −14.33752281740395026135423644313, −12.70803105277211321434391666789, −11.49165617860391645194689454932, −10.29822374363341731523599854168, −7.49674095599262099822723011265, −5.62986612673950982305761112381, −4.85212783589294661663172577240, −2.16305931271196059767504183586, 0,
2.16305931271196059767504183586, 4.85212783589294661663172577240, 5.62986612673950982305761112381, 7.49674095599262099822723011265, 10.29822374363341731523599854168, 11.49165617860391645194689454932, 12.70803105277211321434391666789, 14.33752281740395026135423644313, 15.91083813137086685485187029150
