| L(s) = 1 | + 3-s − 4-s + 9-s − 12-s − 4·13-s − 3·16-s − 6·23-s + 25-s + 27-s − 12·29-s − 36-s − 4·39-s + 10·43-s − 3·48-s − 10·49-s + 4·52-s − 12·53-s − 8·61-s + 7·64-s − 6·69-s + 75-s + 4·79-s + 81-s − 12·87-s + 6·92-s − 100-s + 22·103-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 3/4·16-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 2.22·29-s − 1/6·36-s − 0.640·39-s + 1.52·43-s − 0.433·48-s − 1.42·49-s + 0.554·52-s − 1.64·53-s − 1.02·61-s + 7/8·64-s − 0.722·69-s + 0.115·75-s + 0.450·79-s + 1/9·81-s − 1.28·87-s + 0.625·92-s − 0.0999·100-s + 2.16·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
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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
−9.243883739655684232285086452642, −8.971258763318936071234614278231, −8.116802337879264713575699579331, −7.83442737994774170091406163375, −7.40703833223320554641736294118, −6.82606790356779222452957979984, −6.19209914820645800528883004008, −5.62735221828977959334464775459, −4.93773160626721556914727239569, −4.47284619946525654223380577953, −3.90261330422601089705278577698, −3.22736973636347109410824917240, −2.38001630660605951551548226436, −1.76747923206471972685248272428, 0,
1.76747923206471972685248272428, 2.38001630660605951551548226436, 3.22736973636347109410824917240, 3.90261330422601089705278577698, 4.47284619946525654223380577953, 4.93773160626721556914727239569, 5.62735221828977959334464775459, 6.19209914820645800528883004008, 6.82606790356779222452957979984, 7.40703833223320554641736294118, 7.83442737994774170091406163375, 8.116802337879264713575699579331, 8.971258763318936071234614278231, 9.243883739655684232285086452642
