| L(s) = 1 | − 2·3-s + 9-s + 12·17-s + 6·19-s + 6·25-s + 4·27-s + 12·41-s + 4·43-s − 49-s − 24·51-s − 12·57-s + 6·59-s + 4·67-s − 12·73-s − 12·75-s − 11·81-s + 6·83-s + 24·89-s + 8·97-s − 24·107-s − 12·113-s − 6·121-s − 24·123-s + 127-s − 8·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 2.91·17-s + 1.37·19-s + 6/5·25-s + 0.769·27-s + 1.87·41-s + 0.609·43-s − 1/7·49-s − 3.36·51-s − 1.58·57-s + 0.781·59-s + 0.488·67-s − 1.40·73-s − 1.38·75-s − 1.22·81-s + 0.658·83-s + 2.54·89-s + 0.812·97-s − 2.32·107-s − 1.12·113-s − 0.545·121-s − 2.16·123-s + 0.0887·127-s − 0.704·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.536317381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.536317381\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.566771960485630462019312540254, −7.942520937814782074617937241800, −7.55019412448201410087714654464, −7.36338170740796924148539687423, −6.62300336249697730239508595172, −6.15650359675120277752154275669, −5.69951890082933816205002898423, −5.28658508830661107252117980488, −5.08963413215617845273196562802, −4.33249525321674669091263866004, −3.59885829066484209902594928234, −3.13832780502105678979825059954, −2.54622547375551610623534103893, −1.19300789809051322459157126180, −0.900097405548587559795799958665,
0.900097405548587559795799958665, 1.19300789809051322459157126180, 2.54622547375551610623534103893, 3.13832780502105678979825059954, 3.59885829066484209902594928234, 4.33249525321674669091263866004, 5.08963413215617845273196562802, 5.28658508830661107252117980488, 5.69951890082933816205002898423, 6.15650359675120277752154275669, 6.62300336249697730239508595172, 7.36338170740796924148539687423, 7.55019412448201410087714654464, 7.942520937814782074617937241800, 8.566771960485630462019312540254
