| L(s) = 1 | − 4·13-s + 2·17-s − 6·23-s − 25-s − 12·29-s + 12·43-s + 5·49-s − 10·53-s + 26·61-s − 10·79-s + 28·101-s − 24·103-s + 38·107-s + 12·113-s + 13·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1.10·13-s + 0.485·17-s − 1.25·23-s − 1/5·25-s − 2.22·29-s + 1.82·43-s + 5/7·49-s − 1.37·53-s + 3.32·61-s − 1.12·79-s + 2.78·101-s − 2.36·103-s + 3.67·107-s + 1.12·113-s + 1.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480544396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480544396\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−8.367476474705662297019204128827, −8.092366014125413123960496729716, −7.67726041741918659271991259977, −7.45890809961656071592189114693, −7.10439209355291469261200502734, −6.83817671786102130350292273043, −6.16907301289705674986140884692, −5.88547537334901644791923325136, −5.56749229296028186113629060094, −5.35258383993459473397965795524, −4.57841774673634439232553469131, −4.54818103688673924738484286201, −3.91712739792805610391659398736, −3.55836891004877512095795577162, −3.25107491966868373533959690405, −2.45499252715986305576019236076, −2.19540262175893665987603136453, −1.86645923511347951328213745291, −1.02631436756859149121024453371, −0.36612974206415536403648729625,
0.36612974206415536403648729625, 1.02631436756859149121024453371, 1.86645923511347951328213745291, 2.19540262175893665987603136453, 2.45499252715986305576019236076, 3.25107491966868373533959690405, 3.55836891004877512095795577162, 3.91712739792805610391659398736, 4.54818103688673924738484286201, 4.57841774673634439232553469131, 5.35258383993459473397965795524, 5.56749229296028186113629060094, 5.88547537334901644791923325136, 6.16907301289705674986140884692, 6.83817671786102130350292273043, 7.10439209355291469261200502734, 7.45890809961656071592189114693, 7.67726041741918659271991259977, 8.092366014125413123960496729716, 8.367476474705662297019204128827
