| L(s) = 1 | + 4-s + 3·7-s − 8·13-s + 16-s + 7·19-s + 8·25-s + 3·28-s + 4·31-s − 2·37-s + 43-s − 8·52-s − 5·61-s + 64-s + 28·67-s + 12·73-s + 7·76-s − 14·79-s − 24·91-s + 7·97-s + 8·100-s − 2·103-s − 20·109-s + 3·112-s − 7·121-s + 4·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.13·7-s − 2.21·13-s + 1/4·16-s + 1.60·19-s + 8/5·25-s + 0.566·28-s + 0.718·31-s − 0.328·37-s + 0.152·43-s − 1.10·52-s − 0.640·61-s + 1/8·64-s + 3.42·67-s + 1.40·73-s + 0.802·76-s − 1.57·79-s − 2.51·91-s + 0.710·97-s + 4/5·100-s − 0.197·103-s − 1.91·109-s + 0.283·112-s − 0.636·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.076348791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.076348791\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 2 T + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 11 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−9.315395867058544843961773986566, −8.699528391713256766385124066136, −8.080486028129421915454728815759, −7.79231362179425307530942708107, −7.31136110862664794564680150699, −6.86433893458077974827998245445, −6.47732096389977393293275477001, −5.39352977683038922620478879493, −5.22825377100289241992724167371, −4.82490834230843559592685663185, −4.15813870287003958591964497406, −3.18555535142609381203368985910, −2.67616650614846839500205725771, −2.00623045889002030039972727836, −1.00615658831953362287855636902,
1.00615658831953362287855636902, 2.00623045889002030039972727836, 2.67616650614846839500205725771, 3.18555535142609381203368985910, 4.15813870287003958591964497406, 4.82490834230843559592685663185, 5.22825377100289241992724167371, 5.39352977683038922620478879493, 6.47732096389977393293275477001, 6.86433893458077974827998245445, 7.31136110862664794564680150699, 7.79231362179425307530942708107, 8.080486028129421915454728815759, 8.699528391713256766385124066136, 9.315395867058544843961773986566
