| L(s) = 1 | + 6·4-s + 2·7-s + 32·13-s + 20·16-s + 66·19-s + 18·25-s + 12·28-s + 62·31-s − 114·37-s − 95·49-s + 192·52-s + 210·61-s + 24·64-s − 206·67-s − 94·73-s + 396·76-s + 46·79-s + 64·91-s − 50·97-s + 108·100-s − 50·103-s − 238·109-s + 40·112-s + 372·124-s + 127-s + 131-s + 132·133-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2/7·7-s + 2.46·13-s + 5/4·16-s + 3.47·19-s + 0.719·25-s + 3/7·28-s + 2·31-s − 3.08·37-s − 1.93·49-s + 3.69·52-s + 3.44·61-s + 3/8·64-s − 3.07·67-s − 1.28·73-s + 5.21·76-s + 0.582·79-s + 0.703·91-s − 0.515·97-s + 1.07·100-s − 0.485·103-s − 2.18·109-s + 5/14·112-s + 3·124-s + 0.00787·127-s + 0.00763·131-s + 0.992·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.863020923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.863020923\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T^{2} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 450 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 33 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 480 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1520 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 57 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3120 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 720 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 2320 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2160 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 105 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 103 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 4030 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 47 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 23 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2528 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 13250 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
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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
−10.07420051806461618699947572837, −9.495727103434120399017220993915, −8.989557462170483258618647923452, −8.603718318026404854878091153008, −8.072269743828641353320962854262, −7.957585860392948710862648915010, −7.22142900092143141849208689779, −6.81893965854202978867851421303, −6.79778640389330186747262637349, −5.96069484627692619948180969144, −5.84360145878746044250049397617, −5.15019612548564688930750614090, −4.94733336923083611330269066800, −3.92848988069182232409298318738, −3.52272827202998706707420919678, −2.98984124442600382980158990031, −2.84788741011434866467437108435, −1.60111092638966518630347977692, −1.45650354476959540702847990280, −0.876831941417362337641165149912,
0.876831941417362337641165149912, 1.45650354476959540702847990280, 1.60111092638966518630347977692, 2.84788741011434866467437108435, 2.98984124442600382980158990031, 3.52272827202998706707420919678, 3.92848988069182232409298318738, 4.94733336923083611330269066800, 5.15019612548564688930750614090, 5.84360145878746044250049397617, 5.96069484627692619948180969144, 6.79778640389330186747262637349, 6.81893965854202978867851421303, 7.22142900092143141849208689779, 7.957585860392948710862648915010, 8.072269743828641353320962854262, 8.603718318026404854878091153008, 8.989557462170483258618647923452, 9.495727103434120399017220993915, 10.07420051806461618699947572837
