| L(s) = 1 | + 3-s + 2·5-s − 6·11-s + 6·13-s + 2·15-s + 4·17-s + 5·19-s − 4·23-s + 5·25-s − 27-s − 8·29-s − 7·31-s − 6·33-s + 9·37-s + 6·39-s + 4·41-s + 2·43-s − 2·47-s + 4·51-s − 8·53-s − 12·55-s + 5·57-s + 10·61-s + 12·65-s − 15·67-s − 4·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.80·11-s + 1.66·13-s + 0.516·15-s + 0.970·17-s + 1.14·19-s − 0.834·23-s + 25-s − 0.192·27-s − 1.48·29-s − 1.25·31-s − 1.04·33-s + 1.47·37-s + 0.960·39-s + 0.624·41-s + 0.304·43-s − 0.291·47-s + 0.560·51-s − 1.09·53-s − 1.61·55-s + 0.662·57-s + 1.28·61-s + 1.48·65-s − 1.83·67-s − 0.481·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.569742210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.569742210\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 9 T + 44 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 8 T + 11 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 158 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−9.154484859743414371390450618196, −9.092635781615024831149566403704, −8.205210517210217053071265300186, −8.078145214801767564557446707901, −7.67382145379417477054401446798, −7.48457316933016193796631879025, −6.96868056324058078715678861436, −6.16851003718560532595808192258, −6.09766277252013849112245465211, −5.65153963236873509428823503463, −5.31036434519018779828341791671, −5.02159259287985262831464228901, −4.27524336133453655462096559279, −3.81700844401088334230300828778, −3.21308874414911511995958347128, −3.12047109707466687418107886615, −2.40506266238682761969901650902, −1.94939700911070132198495348400, −1.39298060446458252103929908041, −0.62891033082651639540066656289,
0.62891033082651639540066656289, 1.39298060446458252103929908041, 1.94939700911070132198495348400, 2.40506266238682761969901650902, 3.12047109707466687418107886615, 3.21308874414911511995958347128, 3.81700844401088334230300828778, 4.27524336133453655462096559279, 5.02159259287985262831464228901, 5.31036434519018779828341791671, 5.65153963236873509428823503463, 6.09766277252013849112245465211, 6.16851003718560532595808192258, 6.96868056324058078715678861436, 7.48457316933016193796631879025, 7.67382145379417477054401446798, 8.078145214801767564557446707901, 8.205210517210217053071265300186, 9.092635781615024831149566403704, 9.154484859743414371390450618196
