| L(s) = 1 | − 4-s − 5·7-s + 2·8-s − 2·9-s − 4·11-s + 8·13-s + 16-s − 6·25-s + 5·28-s + 12·31-s − 4·32-s + 2·36-s + 4·41-s + 4·44-s + 14·49-s − 8·52-s − 16·53-s − 10·56-s − 4·61-s + 10·63-s + 3·64-s − 4·72-s + 20·77-s − 5·81-s − 12·83-s − 8·88-s − 40·91-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.88·7-s + 0.707·8-s − 2/3·9-s − 1.20·11-s + 2.21·13-s + 1/4·16-s − 6/5·25-s + 0.944·28-s + 2.15·31-s − 0.707·32-s + 1/3·36-s + 0.624·41-s + 0.603·44-s + 2·49-s − 1.10·52-s − 2.19·53-s − 1.33·56-s − 0.512·61-s + 1.25·63-s + 3/8·64-s − 0.471·72-s + 2.27·77-s − 5/9·81-s − 1.31·83-s − 0.852·88-s − 4.19·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178766 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178766 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 113 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−9.018015913731263211765472540881, −8.343550722748367296659979930604, −8.098331004814877809689653842608, −7.65431991436344816040624452510, −6.86504606060683632411463830259, −6.30708774697536076042957981143, −6.04500050885179815655891288257, −5.62407197070652530477560654444, −4.85764582087153952865463552702, −4.18909485825862815495240690420, −3.63141814836998286628494963738, −3.11946051614226859974258240809, −2.54541598168719447719751135978, −1.25990372511792795659147261413, 0,
1.25990372511792795659147261413, 2.54541598168719447719751135978, 3.11946051614226859974258240809, 3.63141814836998286628494963738, 4.18909485825862815495240690420, 4.85764582087153952865463552702, 5.62407197070652530477560654444, 6.04500050885179815655891288257, 6.30708774697536076042957981143, 6.86504606060683632411463830259, 7.65431991436344816040624452510, 8.098331004814877809689653842608, 8.343550722748367296659979930604, 9.018015913731263211765472540881
