| L(s) = 1 | − 2·2-s + 18·3-s − 124·4-s − 10·5-s − 36·6-s − 902·7-s + 504·8-s − 1.86e3·9-s + 20·10-s − 8.63e3·11-s − 2.23e3·12-s + 1.08e4·13-s + 1.80e3·14-s − 180·15-s + 1.48e4·16-s + 4.91e3·17-s + 3.72e3·18-s − 784·19-s + 1.24e3·20-s − 1.62e4·21-s + 1.72e4·22-s + 7.73e4·23-s + 9.07e3·24-s − 7.80e4·25-s − 2.17e4·26-s − 7.29e4·27-s + 1.11e5·28-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 0.384·3-s − 0.968·4-s − 0.0357·5-s − 0.0680·6-s − 0.993·7-s + 0.348·8-s − 0.851·9-s + 0.00632·10-s − 1.95·11-s − 0.372·12-s + 1.37·13-s + 0.175·14-s − 0.0137·15-s + 0.907·16-s + 0.242·17-s + 0.150·18-s − 0.0262·19-s + 0.0346·20-s − 0.382·21-s + 0.345·22-s + 1.32·23-s + 0.133·24-s − 0.998·25-s − 0.242·26-s − 0.712·27-s + 0.962·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - p^{3} T \) |
| good | 2 | \( 1 + p T + p^{7} T^{2} \) |
| 3 | \( 1 - 2 p^{2} T + p^{7} T^{2} \) |
| 5 | \( 1 + 2 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 902 T + p^{7} T^{2} \) |
| 11 | \( 1 + 8634 T + p^{7} T^{2} \) |
| 13 | \( 1 - 10858 T + p^{7} T^{2} \) |
| 19 | \( 1 + 784 T + p^{7} T^{2} \) |
| 23 | \( 1 - 77330 T + p^{7} T^{2} \) |
| 29 | \( 1 + 18210 T + p^{7} T^{2} \) |
| 31 | \( 1 + 237002 T + p^{7} T^{2} \) |
| 37 | \( 1 - 230878 T + p^{7} T^{2} \) |
| 41 | \( 1 + 304182 T + p^{7} T^{2} \) |
| 43 | \( 1 + 525032 T + p^{7} T^{2} \) |
| 47 | \( 1 - 802752 T + p^{7} T^{2} \) |
| 53 | \( 1 - 152862 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1602408 T + p^{7} T^{2} \) |
| 61 | \( 1 + 2601610 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1074604 T + p^{7} T^{2} \) |
| 71 | \( 1 + 502298 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3648258 T + p^{7} T^{2} \) |
| 79 | \( 1 + 2892174 T + p^{7} T^{2} \) |
| 83 | \( 1 - 728104 T + p^{7} T^{2} \) |
| 89 | \( 1 - 7931846 T + p^{7} T^{2} \) |
| 97 | \( 1 + 6551038 T + p^{7} T^{2} \) |
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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
−16.65260600917003523198692298143, −15.32807681293863038346081192930, −13.63401444969243187477508147830, −12.97828602468115924716208683736, −10.72079702292987871415304628816, −9.238184823413553637238751806062, −8.008474549248749583202771753867, −5.56778791621844211677544881413, −3.27983221456119849599244966498, 0,
3.27983221456119849599244966498, 5.56778791621844211677544881413, 8.008474549248749583202771753867, 9.238184823413553637238751806062, 10.72079702292987871415304628816, 12.97828602468115924716208683736, 13.63401444969243187477508147830, 15.32807681293863038346081192930, 16.65260600917003523198692298143
