| L(s) = 1 | − 46·3-s + 160·5-s − 343·7-s − 71·9-s + 6.84e3·11-s + 2.90e3·13-s − 7.36e3·15-s + 1.65e4·17-s + 6.71e3·19-s + 1.57e4·21-s − 976·23-s − 5.25e4·25-s + 1.03e5·27-s + 6.16e4·29-s − 6.92e4·31-s − 3.14e5·33-s − 5.48e4·35-s + 5.33e5·37-s − 1.33e5·39-s + 1.83e5·41-s − 9.66e5·43-s − 1.13e4·45-s − 1.90e5·47-s + 1.17e5·49-s − 7.62e5·51-s + 7.85e5·53-s + 1.09e6·55-s + ⋯ |
| L(s) = 1 | − 0.983·3-s + 0.572·5-s − 0.377·7-s − 0.0324·9-s + 1.54·11-s + 0.366·13-s − 0.563·15-s + 0.817·17-s + 0.224·19-s + 0.371·21-s − 0.0167·23-s − 0.672·25-s + 1.01·27-s + 0.469·29-s − 0.417·31-s − 1.52·33-s − 0.216·35-s + 1.73·37-s − 0.360·39-s + 0.415·41-s − 1.85·43-s − 0.0185·45-s − 0.267·47-s + 1/7·49-s − 0.804·51-s + 0.724·53-s + 0.886·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.781987098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781987098\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 46 T + p^{7} T^{2} \) |
| 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
−9.881902667865003765926069991450, −9.286284878658147234178431291818, −8.146123849897553523425871336962, −6.80306514462331740666517340857, −6.15825491937290577839344558506, −5.48305352764957676706657730415, −4.25117026984354501742247889295, −3.12299746637901274163340036571, −1.59470626777535399499705163279, −0.66916234109943513666935072490,
0.66916234109943513666935072490, 1.59470626777535399499705163279, 3.12299746637901274163340036571, 4.25117026984354501742247889295, 5.48305352764957676706657730415, 6.15825491937290577839344558506, 6.80306514462331740666517340857, 8.146123849897553523425871336962, 9.286284878658147234178431291818, 9.881902667865003765926069991450
