| L(s) = 1 | − 81·9-s + 432·11-s − 3.16e3·19-s + 9.13e3·29-s + 5.48e3·31-s − 2.67e4·41-s + 3.16e4·49-s + 3.11e4·59-s + 7.86e4·61-s + 1.14e5·71-s + 2.11e4·79-s + 6.56e3·81-s + 2.32e5·89-s − 3.49e4·99-s + 5.52e4·101-s + 4.60e5·109-s − 1.82e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.49e5·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.07·11-s − 2.00·19-s + 2.01·29-s + 1.02·31-s − 2.48·41-s + 1.88·49-s + 1.16·59-s + 2.70·61-s + 2.68·71-s + 0.380·79-s + 1/9·81-s + 3.11·89-s − 0.358·99-s + 0.538·101-s + 3.71·109-s − 1.13·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 0.403·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.129121456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.129121456\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 31678 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 216 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 149686 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2554558 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1580 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4439470 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4566 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2744 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 136608550 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 13350 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 1960730 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 341531038 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 737547622 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 264 p T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 39302 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 412943402 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 57120 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 1605781582 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10552 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 567290 p^{2} T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 116430 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 17166940990 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21115750836384368434329691411, −10.53104170324327817908202198584, −10.17441185710974251919933355535, −9.959430315134498845067613761964, −9.042912236779925476228196944217, −8.768100900965576323745367955383, −8.299413736454956642585167024553, −8.068771507561092723159468306081, −6.93576742330407891727936038341, −6.79751272198149287838526002146, −6.33720193390784163211742803727, −5.80699508266202579623123524944, −4.86707540992403090175228484694, −4.71558616395278519890037109159, −3.73206934808178501428819469534, −3.58330081840341827902651458639, −2.33615532441914227899475679760, −2.21704720789067257294825913843, −1.03548875318616564790633239603, −0.56064957615016467624518410478,
0.56064957615016467624518410478, 1.03548875318616564790633239603, 2.21704720789067257294825913843, 2.33615532441914227899475679760, 3.58330081840341827902651458639, 3.73206934808178501428819469534, 4.71558616395278519890037109159, 4.86707540992403090175228484694, 5.80699508266202579623123524944, 6.33720193390784163211742803727, 6.79751272198149287838526002146, 6.93576742330407891727936038341, 8.068771507561092723159468306081, 8.299413736454956642585167024553, 8.768100900965576323745367955383, 9.042912236779925476228196944217, 9.959430315134498845067613761964, 10.17441185710974251919933355535, 10.53104170324327817908202198584, 11.21115750836384368434329691411
