| L(s) = 1 | − 2·5-s − 4·11-s + 8·19-s + 5·25-s + 14·29-s − 12·31-s + 10·41-s − 13·49-s + 8·55-s − 24·59-s + 14·61-s − 40·71-s − 8·79-s − 60·89-s − 16·95-s + 36·101-s + 36·109-s + 26·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s − 28·145-s + 149-s + 151-s + 24·155-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.20·11-s + 1.83·19-s + 25-s + 2.59·29-s − 2.15·31-s + 1.56·41-s − 1.85·49-s + 1.07·55-s − 3.12·59-s + 1.79·61-s − 4.74·71-s − 0.900·79-s − 6.35·89-s − 1.64·95-s + 3.58·101-s + 3.44·109-s + 2.36·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.32·145-s + 0.0819·149-s + 0.0813·151-s + 1.92·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7835819283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7835819283\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 7 T + 20 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 58 T^{2} + 1515 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 13 T^{2} - 2040 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 141 T^{2} + 12992 T^{4} + 141 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 62 T^{2} - 5565 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.51705240166213152007570652566, −6.34084420252208200709776897714, −5.88839803251556337528240864825, −5.69255090719242987571132975412, −5.61486154709628952937810949744, −5.55741756099638326030186871949, −5.21595294550443145040599821086, −4.87665054025083822360361555737, −4.59672595427736479484994834085, −4.59132994242869465085694220618, −4.32970464568374621884797804744, −4.31471640458441348609211874951, −3.85805029406410296845859215083, −3.38112363135966990741075046991, −3.33287295139578123854568996966, −3.10192491135609643412026370188, −2.93147751856421482378889068410, −2.80542572569371836071911928753, −2.48806038583656296070306932190, −2.00374971495860340207095191435, −1.73697515878946547055340031066, −1.31427732674723225173679902676, −1.21524416204542708796084851731, −0.66752585687098120215768822554, −0.18660954820541349242208180974,
0.18660954820541349242208180974, 0.66752585687098120215768822554, 1.21524416204542708796084851731, 1.31427732674723225173679902676, 1.73697515878946547055340031066, 2.00374971495860340207095191435, 2.48806038583656296070306932190, 2.80542572569371836071911928753, 2.93147751856421482378889068410, 3.10192491135609643412026370188, 3.33287295139578123854568996966, 3.38112363135966990741075046991, 3.85805029406410296845859215083, 4.31471640458441348609211874951, 4.32970464568374621884797804744, 4.59132994242869465085694220618, 4.59672595427736479484994834085, 4.87665054025083822360361555737, 5.21595294550443145040599821086, 5.55741756099638326030186871949, 5.61486154709628952937810949744, 5.69255090719242987571132975412, 5.88839803251556337528240864825, 6.34084420252208200709776897714, 6.51705240166213152007570652566
