| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 6·9-s + 5·16-s − 12·18-s + 16·23-s − 6·25-s + 4·29-s + 6·32-s − 18·36-s + 20·37-s + 16·43-s + 32·46-s − 7·49-s − 12·50-s − 12·53-s + 8·58-s + 7·64-s − 24·67-s + 16·71-s − 24·72-s + 40·74-s − 16·79-s + 27·81-s + 32·86-s + 48·92-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s + 5/4·16-s − 2.82·18-s + 3.33·23-s − 6/5·25-s + 0.742·29-s + 1.06·32-s − 3·36-s + 3.28·37-s + 2.43·43-s + 4.71·46-s − 49-s − 1.69·50-s − 1.64·53-s + 1.05·58-s + 7/8·64-s − 2.93·67-s + 1.89·71-s − 2.82·72-s + 4.64·74-s − 1.80·79-s + 3·81-s + 3.45·86-s + 5.00·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188356 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188356 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.753323965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.753323965\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.056498267458590117851162621111, −8.767717911672973124583558991985, −7.85520408569034144174211600673, −7.76857292337562945056886541715, −7.13164735275382460400374069877, −6.32858579420662158347624019611, −6.10481334469888763444585228494, −5.73966246913035569800644487489, −4.99493437014528324856589187903, −4.74203912193695697715499891246, −4.05486645798658988472696574741, −3.16188554299090604324575557253, −2.88035578683009317838785569193, −2.44397171990298187609757473706, −1.04640991976554111715719591186,
1.04640991976554111715719591186, 2.44397171990298187609757473706, 2.88035578683009317838785569193, 3.16188554299090604324575557253, 4.05486645798658988472696574741, 4.74203912193695697715499891246, 4.99493437014528324856589187903, 5.73966246913035569800644487489, 6.10481334469888763444585228494, 6.32858579420662158347624019611, 7.13164735275382460400374069877, 7.76857292337562945056886541715, 7.85520408569034144174211600673, 8.767717911672973124583558991985, 9.056498267458590117851162621111
