| L(s) = 1 | − 2-s − 2·3-s − 2·4-s + 2·6-s − 6·7-s + 3·8-s + 3·9-s + 4·12-s + 2·13-s + 6·14-s + 16-s + 4·17-s − 3·18-s + 10·19-s + 12·21-s + 7·23-s − 6·24-s − 2·26-s − 4·27-s + 12·28-s − 5·29-s + 14·31-s − 2·32-s − 4·34-s − 6·36-s − 6·37-s − 10·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 4-s + 0.816·6-s − 2.26·7-s + 1.06·8-s + 9-s + 1.15·12-s + 0.554·13-s + 1.60·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 2.29·19-s + 2.61·21-s + 1.45·23-s − 1.22·24-s − 0.392·26-s − 0.769·27-s + 2.26·28-s − 0.928·29-s + 2.51·31-s − 0.353·32-s − 0.685·34-s − 36-s − 0.986·37-s − 1.62·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8818341658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8818341658\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 53 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 14 T + 111 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + 63 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 147 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 73 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 21 T + 243 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 207 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 133 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15 T + 223 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 223 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.57368825495617663990819461557, −7.57092483065320991340095813349, −7.26118850252987331100961143908, −7.07300950601451656825428542510, −6.30763414854500950073895718061, −6.05911989980354386544903327572, −6.05564317611295830637091442587, −5.68172436366251250175226911986, −5.06407135204977400579289229628, −4.89561538569563748015592263659, −4.45558256406378144687825595793, −4.14767042175501508488456261773, −3.43102708359153233087557833190, −3.35270688127939944859521606360, −2.99209793792457470883050491763, −2.61930875908172074648985185154, −1.57067284450317578710555426985, −1.12106752524639717251524380318, −0.63793014732322263644745477697, −0.52997886581542145951744092945,
0.52997886581542145951744092945, 0.63793014732322263644745477697, 1.12106752524639717251524380318, 1.57067284450317578710555426985, 2.61930875908172074648985185154, 2.99209793792457470883050491763, 3.35270688127939944859521606360, 3.43102708359153233087557833190, 4.14767042175501508488456261773, 4.45558256406378144687825595793, 4.89561538569563748015592263659, 5.06407135204977400579289229628, 5.68172436366251250175226911986, 6.05564317611295830637091442587, 6.05911989980354386544903327572, 6.30763414854500950073895718061, 7.07300950601451656825428542510, 7.26118850252987331100961143908, 7.57092483065320991340095813349, 7.57368825495617663990819461557
