L(s) = 1 | − 2-s + 4-s − 8-s + 9-s + 2·11-s + 16-s + 2·17-s − 18-s − 2·19-s − 2·22-s + 2·25-s − 32-s − 2·34-s + 36-s + 2·38-s + 18·41-s + 2·44-s − 4·49-s − 2·50-s + 10·59-s + 64-s − 10·67-s + 2·68-s − 72-s + 10·73-s − 2·76-s + 81-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s − 0.426·22-s + 2/5·25-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.324·38-s + 2.81·41-s + 0.301·44-s − 4/7·49-s − 0.282·50-s + 1.30·59-s + 1/8·64-s − 1.22·67-s + 0.242·68-s − 0.117·72-s + 1.17·73-s − 0.229·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 332928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.390815689\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.390815689\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 80 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 140 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.871747591477723490966823796019, −8.284219306744146302033965153673, −7.83326095338794487804025198121, −7.51589596344692985995331997935, −6.88582590126517908164325867497, −6.57036829620668455053553584045, −5.99377669881420947816614031366, −5.56651040385884547633307572089, −4.89175420348459165822884119569, −4.20896574320434341938862312034, −3.85107108870254959267147611793, −3.01000289628689571131451698315, −2.43876237620700989917098389577, −1.60437373389557375530623529932, −0.801294814325978538714061369619,
0.801294814325978538714061369619, 1.60437373389557375530623529932, 2.43876237620700989917098389577, 3.01000289628689571131451698315, 3.85107108870254959267147611793, 4.20896574320434341938862312034, 4.89175420348459165822884119569, 5.56651040385884547633307572089, 5.99377669881420947816614031366, 6.57036829620668455053553584045, 6.88582590126517908164325867497, 7.51589596344692985995331997935, 7.83326095338794487804025198121, 8.284219306744146302033965153673, 8.871747591477723490966823796019