| L(s) = 1 | + 2·3-s + 7-s + 9-s − 2·13-s − 12·19-s + 2·21-s − 2·25-s − 4·27-s + 8·31-s + 12·37-s − 4·39-s + 12·43-s + 49-s − 24·57-s + 6·61-s + 63-s − 4·67-s − 6·73-s − 4·75-s + 20·79-s − 11·81-s − 2·91-s + 16·93-s + 10·97-s + 8·103-s + 16·109-s + 24·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 0.554·13-s − 2.75·19-s + 0.436·21-s − 2/5·25-s − 0.769·27-s + 1.43·31-s + 1.97·37-s − 0.640·39-s + 1.82·43-s + 1/7·49-s − 3.17·57-s + 0.768·61-s + 0.125·63-s − 0.488·67-s − 0.702·73-s − 0.461·75-s + 2.25·79-s − 1.22·81-s − 0.209·91-s + 1.65·93-s + 1.01·97-s + 0.788·103-s + 1.53·109-s + 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790272 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.645803405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645803405\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
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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
−8.195705641190379758431729992077, −7.88099402769952280940767920521, −7.65122014077856182391313130630, −6.94320533701928241195249314144, −6.52058097001450308203999629314, −5.93055967326737445249277323884, −5.78231564479682787522682773585, −4.67698821723675326605841826787, −4.54541820871378036201069737886, −4.08091559282306906227885055933, −3.46596176490541165827210269303, −2.71475289051191024014619496909, −2.30569624925039573556900502847, −1.96696703718302186666602557851, −0.71262587750700779146121225580,
0.71262587750700779146121225580, 1.96696703718302186666602557851, 2.30569624925039573556900502847, 2.71475289051191024014619496909, 3.46596176490541165827210269303, 4.08091559282306906227885055933, 4.54541820871378036201069737886, 4.67698821723675326605841826787, 5.78231564479682787522682773585, 5.93055967326737445249277323884, 6.52058097001450308203999629314, 6.94320533701928241195249314144, 7.65122014077856182391313130630, 7.88099402769952280940767920521, 8.195705641190379758431729992077
