| L(s) = 1 | + 2·7-s − 2·13-s + 19-s + 2·25-s + 31-s − 2·37-s + 43-s + 3·49-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s + 97-s + 4·103-s + 109-s + 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 2·7-s − 2·13-s + 19-s + 2·25-s + 31-s − 2·37-s + 43-s + 3·49-s + 61-s − 2·67-s + 73-s − 2·79-s − 4·91-s + 97-s + 4·103-s + 109-s + 2·121-s + 127-s + 131-s + 2·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.632717920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632717920\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.188754919180011856069537541088, −8.965197137639307565490345864460, −8.611689903288101238030312326419, −8.359206202659616191092218338091, −7.64176372835243648768349244139, −7.57190484254547160774300182641, −7.10889829460470406674706521038, −7.04600569080584849253110280983, −6.22124977003612519667269099114, −5.81265083752014871343383155148, −5.11748946445603479251160522865, −5.10522681633471481915527360708, −4.62358964249994656542382993997, −4.53575872427349756876920266939, −3.65645765918656192194199195432, −3.18668971611496039336755154062, −2.42912079975785856482785934256, −2.35274301910678028761151887825, −1.49557311344948550455527650027, −0.982146521603076594537823915266,
0.982146521603076594537823915266, 1.49557311344948550455527650027, 2.35274301910678028761151887825, 2.42912079975785856482785934256, 3.18668971611496039336755154062, 3.65645765918656192194199195432, 4.53575872427349756876920266939, 4.62358964249994656542382993997, 5.10522681633471481915527360708, 5.11748946445603479251160522865, 5.81265083752014871343383155148, 6.22124977003612519667269099114, 7.04600569080584849253110280983, 7.10889829460470406674706521038, 7.57190484254547160774300182641, 7.64176372835243648768349244139, 8.359206202659616191092218338091, 8.611689903288101238030312326419, 8.965197137639307565490345864460, 9.188754919180011856069537541088
