| L(s) = 1 | + 160·5-s − 343·7-s + 6.84e3·11-s − 2.90e3·13-s − 1.65e4·17-s − 6.71e3·19-s + 976·23-s − 5.25e4·25-s + 6.16e4·29-s − 6.92e4·31-s − 5.48e4·35-s − 5.33e5·37-s − 1.83e5·41-s + 9.66e5·43-s + 1.90e5·47-s + 1.17e5·49-s + 7.85e5·53-s + 1.09e6·55-s − 2.89e6·59-s − 9.58e4·61-s − 4.64e5·65-s − 9.91e5·67-s − 1.06e6·71-s + 2.52e6·73-s − 2.34e6·77-s + 2.85e5·79-s − 7.09e6·83-s + ⋯ |
| L(s) = 1 | + 0.572·5-s − 0.377·7-s + 1.54·11-s − 0.366·13-s − 0.817·17-s − 0.224·19-s + 0.0167·23-s − 0.672·25-s + 0.469·29-s − 0.417·31-s − 0.216·35-s − 1.73·37-s − 0.415·41-s + 1.85·43-s + 0.267·47-s + 1/7·49-s + 0.724·53-s + 0.886·55-s − 1.83·59-s − 0.0540·61-s − 0.209·65-s − 0.402·67-s − 0.354·71-s + 0.759·73-s − 0.585·77-s + 0.0652·79-s − 1.36·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 5 | \( 1 - 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 6840 T + p^{7} T^{2} \) |
| 13 | \( 1 + 2900 T + p^{7} T^{2} \) |
| 17 | \( 1 + 16566 T + p^{7} T^{2} \) |
| 19 | \( 1 + 6718 T + p^{7} T^{2} \) |
| 23 | \( 1 - 976 T + p^{7} T^{2} \) |
| 29 | \( 1 - 61662 T + p^{7} T^{2} \) |
| 31 | \( 1 + 69236 T + p^{7} T^{2} \) |
| 37 | \( 1 + 533062 T + p^{7} T^{2} \) |
| 41 | \( 1 + 183158 T + p^{7} T^{2} \) |
| 43 | \( 1 - 966864 T + p^{7} T^{2} \) |
| 47 | \( 1 - 190268 T + p^{7} T^{2} \) |
| 53 | \( 1 - 785010 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2893594 T + p^{7} T^{2} \) |
| 61 | \( 1 + 95896 T + p^{7} T^{2} \) |
| 67 | \( 1 + 991644 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1068160 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2523458 T + p^{7} T^{2} \) |
| 79 | \( 1 - 285848 T + p^{7} T^{2} \) |
| 83 | \( 1 + 7094938 T + p^{7} T^{2} \) |
| 89 | \( 1 - 252390 T + p^{7} T^{2} \) |
| 97 | \( 1 + 1824794 T + p^{7} T^{2} \) |
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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
−9.258613466510256434370301669858, −8.733258954856603251660315078854, −7.33697557130672408377368667970, −6.53872848419034886629005382822, −5.78166006549148570393319928509, −4.52099969654228695501655265317, −3.59951770344855555979841821508, −2.30583993716523161450841664353, −1.33801742610716765808418334057, 0,
1.33801742610716765808418334057, 2.30583993716523161450841664353, 3.59951770344855555979841821508, 4.52099969654228695501655265317, 5.78166006549148570393319928509, 6.53872848419034886629005382822, 7.33697557130672408377368667970, 8.733258954856603251660315078854, 9.258613466510256434370301669858
