| L(s) = 1 | + 4-s + 9-s + 2·11-s − 6·19-s + 32·29-s + 4·31-s + 36-s − 44·41-s + 2·44-s − 2·49-s + 8·59-s − 64-s − 24·71-s − 6·76-s − 16·79-s − 20·89-s + 2·99-s + 12·109-s + 32·116-s + 23·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 0.603·11-s − 1.37·19-s + 5.94·29-s + 0.718·31-s + 1/6·36-s − 6.87·41-s + 0.301·44-s − 2/7·49-s + 1.04·59-s − 1/8·64-s − 2.84·71-s − 0.688·76-s − 1.80·79-s − 2.11·89-s + 0.201·99-s + 1.14·109-s + 2.97·116-s + 2.09·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.633626001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.633626001\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 69 T^{2} + 2552 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 15 T^{2} - 2584 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.10673217437251471965409218023, −6.75345232711337025105040038236, −6.61626681642642048571109342739, −6.51719435760704275130596534428, −6.27247098058301995790817486838, −6.25656280712652071989658912377, −5.75560105492558559235404173295, −5.61029690597784776309569791196, −5.06810510057466644561692028371, −4.95534410521548148420021826264, −4.71165555941424503048609533944, −4.63722305760108412589690753785, −4.55530121253917408076947501295, −3.96160036411659492744353436473, −3.78484761449542754336819170611, −3.54871004938197672819202630355, −3.15955263528094768091559034885, −2.81298008647284192148154497594, −2.76287387021183708056156735574, −2.54925699733949294482816926016, −1.93264561848697945462907098021, −1.65171150572162735313917037641, −1.26861404259426359437036417380, −1.18596051240309613697934638312, −0.27335889485403576172885470641,
0.27335889485403576172885470641, 1.18596051240309613697934638312, 1.26861404259426359437036417380, 1.65171150572162735313917037641, 1.93264561848697945462907098021, 2.54925699733949294482816926016, 2.76287387021183708056156735574, 2.81298008647284192148154497594, 3.15955263528094768091559034885, 3.54871004938197672819202630355, 3.78484761449542754336819170611, 3.96160036411659492744353436473, 4.55530121253917408076947501295, 4.63722305760108412589690753785, 4.71165555941424503048609533944, 4.95534410521548148420021826264, 5.06810510057466644561692028371, 5.61029690597784776309569791196, 5.75560105492558559235404173295, 6.25656280712652071989658912377, 6.27247098058301995790817486838, 6.51719435760704275130596534428, 6.61626681642642048571109342739, 6.75345232711337025105040038236, 7.10673217437251471965409218023
