| L(s) = 1 | + 5-s − 2·9-s + 8·13-s − 2·25-s + 8·29-s − 4·37-s − 4·41-s − 2·45-s − 6·49-s + 12·61-s + 8·65-s + 16·73-s − 5·81-s + 12·89-s − 16·97-s + 16·101-s − 12·109-s + 4·113-s − 16·117-s + 10·121-s − 10·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 2/3·9-s + 2.21·13-s − 2/5·25-s + 1.48·29-s − 0.657·37-s − 0.624·41-s − 0.298·45-s − 6/7·49-s + 1.53·61-s + 0.992·65-s + 1.87·73-s − 5/9·81-s + 1.27·89-s − 1.62·97-s + 1.59·101-s − 1.14·109-s + 0.376·113-s − 1.47·117-s + 0.909·121-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476065523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476065523\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.23483205813984290225818496300, −9.853978785687170244847947278879, −9.055352796670091871403837256883, −8.736065460510876733536467394741, −8.236423344838777274327900802027, −7.87404095627341833844541295753, −6.78487421378196083885900723178, −6.52671736717948083218171716455, −5.92619491824543261132073685088, −5.42058580757221008274658928533, −4.70994800099241165774839580607, −3.77394723045244055985457001783, −3.33265646990171412767986454438, −2.32819754727063863592932131080, −1.23201411688396714069639811312,
1.23201411688396714069639811312, 2.32819754727063863592932131080, 3.33265646990171412767986454438, 3.77394723045244055985457001783, 4.70994800099241165774839580607, 5.42058580757221008274658928533, 5.92619491824543261132073685088, 6.52671736717948083218171716455, 6.78487421378196083885900723178, 7.87404095627341833844541295753, 8.236423344838777274327900802027, 8.736065460510876733536467394741, 9.055352796670091871403837256883, 9.853978785687170244847947278879, 10.23483205813984290225818496300
