| L(s) = 1 | − 3-s − 2·4-s + 9-s + 2·12-s − 3·13-s − 3·19-s + 7·25-s − 27-s − 7·31-s − 2·36-s − 11·37-s + 3·39-s − 10·43-s − 6·49-s + 6·52-s + 3·57-s + 2·61-s + 8·64-s − 3·67-s − 8·73-s − 7·75-s + 6·76-s − 8·79-s + 81-s + 7·93-s − 12·97-s − 14·100-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 1/3·9-s + 0.577·12-s − 0.832·13-s − 0.688·19-s + 7/5·25-s − 0.192·27-s − 1.25·31-s − 1/3·36-s − 1.80·37-s + 0.480·39-s − 1.52·43-s − 6/7·49-s + 0.832·52-s + 0.397·57-s + 0.256·61-s + 64-s − 0.366·67-s − 0.936·73-s − 0.808·75-s + 0.688·76-s − 0.900·79-s + 1/9·81-s + 0.725·93-s − 1.21·97-s − 7/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30807 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30807 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 163 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 24 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 64 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−10.21309792233837538405495649427, −9.733095670555278034616774183418, −9.203899433076560942083504984608, −8.614747220925517039366359706047, −8.364182883445352190391358475614, −7.39792618655523881989646148716, −6.96075190309575119961518791442, −6.45736203566885958940739555265, −5.53206408016868337095675594641, −5.07011362181163262904451136757, −4.61647191494274960730300305256, −3.88841003603198616812608673478, −3.01745522248374523198408954426, −1.77123594625188930046882574507, 0,
1.77123594625188930046882574507, 3.01745522248374523198408954426, 3.88841003603198616812608673478, 4.61647191494274960730300305256, 5.07011362181163262904451136757, 5.53206408016868337095675594641, 6.45736203566885958940739555265, 6.96075190309575119961518791442, 7.39792618655523881989646148716, 8.364182883445352190391358475614, 8.614747220925517039366359706047, 9.203899433076560942083504984608, 9.733095670555278034616774183418, 10.21309792233837538405495649427
