| L(s) = 1 | − 3-s − 2·9-s + 7·11-s − 2·17-s + 11·19-s − 25-s + 5·27-s − 7·33-s + 3·41-s + 22·43-s + 4·49-s + 2·51-s − 11·57-s + 3·59-s − 2·67-s − 5·73-s + 75-s + 81-s + 5·83-s − 31·89-s − 14·99-s + 33·107-s + 24·113-s + 17·121-s − 3·123-s + 127-s − 22·129-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 2.11·11-s − 0.485·17-s + 2.52·19-s − 1/5·25-s + 0.962·27-s − 1.21·33-s + 0.468·41-s + 3.35·43-s + 4/7·49-s + 0.280·51-s − 1.45·57-s + 0.390·59-s − 0.244·67-s − 0.585·73-s + 0.115·75-s + 1/9·81-s + 0.548·83-s − 3.28·89-s − 1.40·99-s + 3.19·107-s + 2.25·113-s + 1.54·121-s − 0.270·123-s + 0.0887·127-s − 1.93·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.197315467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.197315467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 101 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 14 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−7.84369112465311388294969193356, −7.40609340104798936809930105801, −7.13673602625143976386966872870, −6.67436346744420210570206683815, −6.06617722808882638955131407050, −5.83413174558154237589899256942, −5.51363842933161484669790251813, −4.86164218842334300295984666375, −4.28523885229310516214982475768, −3.98213986253367036980968321951, −3.31122350514674822264105862375, −2.87734486943569574231229640332, −2.13370940718258320912760339068, −1.18466849229826933639655668958, −0.817336431668422018547058061540,
0.817336431668422018547058061540, 1.18466849229826933639655668958, 2.13370940718258320912760339068, 2.87734486943569574231229640332, 3.31122350514674822264105862375, 3.98213986253367036980968321951, 4.28523885229310516214982475768, 4.86164218842334300295984666375, 5.51363842933161484669790251813, 5.83413174558154237589899256942, 6.06617722808882638955131407050, 6.67436346744420210570206683815, 7.13673602625143976386966872870, 7.40609340104798936809930105801, 7.84369112465311388294969193356
