| L(s) = 1 | + 2-s + 2·4-s + 5-s + 5·8-s + 10-s + 5·11-s − 5·13-s + 5·16-s + 6·17-s − 2·19-s + 2·20-s + 5·22-s + 3·23-s + 5·25-s − 5·26-s − 29-s + 10·32-s + 6·34-s + 6·37-s − 2·38-s + 5·40-s + 5·41-s + 43-s + 10·44-s + 3·46-s + 5·50-s − 10·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.447·5-s + 1.76·8-s + 0.316·10-s + 1.50·11-s − 1.38·13-s + 5/4·16-s + 1.45·17-s − 0.458·19-s + 0.447·20-s + 1.06·22-s + 0.625·23-s + 25-s − 0.980·26-s − 0.185·29-s + 1.76·32-s + 1.02·34-s + 0.986·37-s − 0.324·38-s + 0.790·40-s + 0.780·41-s + 0.152·43-s + 1.50·44-s + 0.442·46-s + 0.707·50-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.316075431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.316075431\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−9.765654098758353579689589013217, −9.586589070570195209519435301502, −9.120325512982210974524711563501, −8.660790446321989972035890209934, −7.917792923176754336091702883245, −7.83251739516913704350720473763, −7.14676001407335559451244264342, −7.02564641711679470513237426715, −6.57975159154352760199619381254, −6.16084346737969298268377596778, −5.49820462400345034607263816226, −5.31943681330000463004881059574, −4.73996282288709150474556684470, −4.12162831739130975987444954383, −4.06224825622680534074913512773, −3.20436812250029263619432161122, −2.62445024600094657434524034121, −2.29332353966590750384064075843, −1.37531438124214562099809327447, −1.09410855423417378425821570018,
1.09410855423417378425821570018, 1.37531438124214562099809327447, 2.29332353966590750384064075843, 2.62445024600094657434524034121, 3.20436812250029263619432161122, 4.06224825622680534074913512773, 4.12162831739130975987444954383, 4.73996282288709150474556684470, 5.31943681330000463004881059574, 5.49820462400345034607263816226, 6.16084346737969298268377596778, 6.57975159154352760199619381254, 7.02564641711679470513237426715, 7.14676001407335559451244264342, 7.83251739516913704350720473763, 7.917792923176754336091702883245, 8.660790446321989972035890209934, 9.120325512982210974524711563501, 9.586589070570195209519435301502, 9.765654098758353579689589013217
