| L(s) = 1 | + 2·2-s + 2·4-s + 7-s − 4·9-s + 2·14-s − 4·16-s + 17-s − 8·18-s − 2·23-s − 4·25-s + 2·28-s + 6·31-s − 8·32-s + 2·34-s − 8·36-s + 12·41-s − 4·46-s + 2·47-s − 6·49-s − 8·50-s + 12·62-s − 4·63-s − 8·64-s + 2·68-s + 4·71-s − 10·73-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 4/3·9-s + 0.534·14-s − 16-s + 0.242·17-s − 1.88·18-s − 0.417·23-s − 4/5·25-s + 0.377·28-s + 1.07·31-s − 1.41·32-s + 0.342·34-s − 4/3·36-s + 1.87·41-s − 0.589·46-s + 0.291·47-s − 6/7·49-s − 1.13·50-s + 1.52·62-s − 0.503·63-s − 64-s + 0.242·68-s + 0.474·71-s − 1.17·73-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.665648815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.665648815\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.86796454325680324201247329048, −11.33809450153631846155985068442, −11.11890383617850006031693317343, −10.20764449160937497348041779737, −9.505464447729603117554406796145, −8.864180222697906748921269445953, −8.206829440803985985373018251480, −7.65237878488682083370291266879, −6.69684744283997543774024577874, −5.96014369474091064082098996797, −5.65759393439021059427373926377, −4.81010255527792915896859759276, −4.11341095066117908255991268990, −3.19739992941210183822018067047, −2.38396094192538377492616969522,
2.38396094192538377492616969522, 3.19739992941210183822018067047, 4.11341095066117908255991268990, 4.81010255527792915896859759276, 5.65759393439021059427373926377, 5.96014369474091064082098996797, 6.69684744283997543774024577874, 7.65237878488682083370291266879, 8.206829440803985985373018251480, 8.864180222697906748921269445953, 9.505464447729603117554406796145, 10.20764449160937497348041779737, 11.11890383617850006031693317343, 11.33809450153631846155985068442, 11.86796454325680324201247329048
