| L(s) = 1 | + 4-s + 4·7-s + 4·13-s + 16-s − 2·19-s − 10·25-s + 4·28-s − 8·31-s − 8·37-s − 2·43-s − 2·49-s + 4·52-s + 16·61-s + 64-s + 10·67-s + 22·73-s − 2·76-s − 8·79-s + 16·91-s + 10·97-s − 10·100-s + 28·103-s − 32·109-s + 4·112-s − 13·121-s − 8·124-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s + 1.10·13-s + 1/4·16-s − 0.458·19-s − 2·25-s + 0.755·28-s − 1.43·31-s − 1.31·37-s − 0.304·43-s − 2/7·49-s + 0.554·52-s + 2.04·61-s + 1/8·64-s + 1.22·67-s + 2.57·73-s − 0.229·76-s − 0.900·79-s + 1.67·91-s + 1.01·97-s − 100-s + 2.75·103-s − 3.06·109-s + 0.377·112-s − 1.18·121-s − 0.718·124-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.572084960\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572084960\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.96157598355786499811507828582, −10.11405722912123694469504278059, −9.711604924732818951217103858923, −8.789487803962222669402730529275, −8.543319215757720981308384286559, −7.80670854475898716623947697687, −7.63671497466111343831458484773, −6.68619713403218599725876916726, −6.25936593777468405983226275094, −5.34383976963704470462884839275, −5.13805326110939411578475614907, −3.94847330019587176084970982031, −3.65076060485342096132488694617, −2.19879385611801234222671009191, −1.60693251106342657124988453680,
1.60693251106342657124988453680, 2.19879385611801234222671009191, 3.65076060485342096132488694617, 3.94847330019587176084970982031, 5.13805326110939411578475614907, 5.34383976963704470462884839275, 6.25936593777468405983226275094, 6.68619713403218599725876916726, 7.63671497466111343831458484773, 7.80670854475898716623947697687, 8.543319215757720981308384286559, 8.789487803962222669402730529275, 9.711604924732818951217103858923, 10.11405722912123694469504278059, 10.96157598355786499811507828582
