| L(s) = 1 | − 2-s − 4-s − 8·7-s + 3·8-s − 9-s + 8·14-s − 16-s + 6·17-s + 18-s − 8·23-s + 8·28-s + 8·31-s − 5·32-s − 6·34-s + 36-s − 10·41-s + 8·46-s + 16·47-s + 34·49-s − 24·56-s − 8·62-s + 8·63-s + 7·64-s − 6·68-s + 16·71-s − 3·72-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 3.02·7-s + 1.06·8-s − 1/3·9-s + 2.13·14-s − 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.66·23-s + 1.51·28-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/6·36-s − 1.56·41-s + 1.17·46-s + 2.33·47-s + 34/7·49-s − 3.20·56-s − 1.01·62-s + 1.00·63-s + 7/8·64-s − 0.727·68-s + 1.89·71-s − 0.353·72-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4003220465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4003220465\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 103 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−13.23784010218841081582418256214, −12.27663322469443509525454591162, −12.05585048460620626423862543456, −11.08447538029037148799030712718, −10.34197819489543942038129863563, −9.959362729867815666331769878362, −9.948425133667698485563105980070, −9.347067482071268746374524742045, −8.937468376288715202210951689621, −8.148204716509848239891592748303, −7.893476631881274387086974573659, −6.88185987515442619539598265197, −6.75150824391477632329555764682, −5.79093089168755001397541133467, −5.73191656978605985197092450600, −4.54698847704968478810073318200, −3.58908890321402914729492351441, −3.45433537376310144986067606263, −2.43013874299533290114921261546, −0.62036380948190494547338357103,
0.62036380948190494547338357103, 2.43013874299533290114921261546, 3.45433537376310144986067606263, 3.58908890321402914729492351441, 4.54698847704968478810073318200, 5.73191656978605985197092450600, 5.79093089168755001397541133467, 6.75150824391477632329555764682, 6.88185987515442619539598265197, 7.893476631881274387086974573659, 8.148204716509848239891592748303, 8.937468376288715202210951689621, 9.347067482071268746374524742045, 9.948425133667698485563105980070, 9.959362729867815666331769878362, 10.34197819489543942038129863563, 11.08447538029037148799030712718, 12.05585048460620626423862543456, 12.27663322469443509525454591162, 13.23784010218841081582418256214
