| L(s) = 1 | + 4·7-s − 4·19-s − 2·25-s − 12·29-s + 16·31-s − 4·37-s + 9·49-s − 12·53-s + 12·59-s − 12·83-s − 8·103-s + 28·109-s + 36·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s − 8·175-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.917·19-s − 2/5·25-s − 2.22·29-s + 2.87·31-s − 0.657·37-s + 9/7·49-s − 1.64·53-s + 1.56·59-s − 1.31·83-s − 0.788·103-s + 2.68·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + 1.07·169-s + 0.0760·173-s − 0.604·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.299663072\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.299663072\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.17717654675375195319275814981, −9.676006291914819441269344034236, −9.538757810048670710223571516680, −8.601687696210814962771173394443, −8.443688551158381539660317915026, −8.380163627580030627585190924310, −7.51192819520089582143565234675, −7.47724639276814183680221468456, −6.87160452710330688023772681563, −6.24622688126620880327132712726, −5.91276184414972441064452187670, −5.43450964107125108163893723856, −4.74257174425157228390661136450, −4.62942892014019535409008522987, −4.05998227547414099170188174269, −3.45392698246985011461660649199, −2.78389696342066649299941617996, −1.97226932497993461772976755695, −1.76129286182797138282354903230, −0.70507083385500222514702759654,
0.70507083385500222514702759654, 1.76129286182797138282354903230, 1.97226932497993461772976755695, 2.78389696342066649299941617996, 3.45392698246985011461660649199, 4.05998227547414099170188174269, 4.62942892014019535409008522987, 4.74257174425157228390661136450, 5.43450964107125108163893723856, 5.91276184414972441064452187670, 6.24622688126620880327132712726, 6.87160452710330688023772681563, 7.47724639276814183680221468456, 7.51192819520089582143565234675, 8.380163627580030627585190924310, 8.443688551158381539660317915026, 8.601687696210814962771173394443, 9.538757810048670710223571516680, 9.676006291914819441269344034236, 10.17717654675375195319275814981
