| L(s) = 1 | + 6·9-s − 4·11-s − 12·19-s − 5·25-s − 4·41-s − 6·49-s + 28·59-s + 27·81-s + 28·89-s − 24·99-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6·169-s − 72·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.20·11-s − 2.75·19-s − 25-s − 0.624·41-s − 6/7·49-s + 3.64·59-s + 3·81-s + 2.96·89-s − 2.41·99-s − 0.909·121-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 + 6/13·169-s − 5.50·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.646847634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.646847634\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−10.01189834274754399308534160477, −9.508218434809640462552057689155, −9.154939644273593689723852753604, −8.416304929789343801293118197581, −8.389866350791385063712407009806, −7.74592460986006238385529971400, −7.55725113131992085501945558784, −6.93648865182708136909338145864, −6.51323186750480796166338039502, −6.42293634005169415510828363066, −5.64325386138238628445848631646, −5.13170956332301373270710468705, −4.78717868140838772970687718482, −4.12362414051404373152275425894, −4.02991545464335586041913743014, −3.42512443476988478287900229256, −2.43094823568408556388572935045, −2.15052945118002727801548050054, −1.61534907303435197837576917532, −0.52902945122450277243009553487,
0.52902945122450277243009553487, 1.61534907303435197837576917532, 2.15052945118002727801548050054, 2.43094823568408556388572935045, 3.42512443476988478287900229256, 4.02991545464335586041913743014, 4.12362414051404373152275425894, 4.78717868140838772970687718482, 5.13170956332301373270710468705, 5.64325386138238628445848631646, 6.42293634005169415510828363066, 6.51323186750480796166338039502, 6.93648865182708136909338145864, 7.55725113131992085501945558784, 7.74592460986006238385529971400, 8.389866350791385063712407009806, 8.416304929789343801293118197581, 9.154939644273593689723852753604, 9.508218434809640462552057689155, 10.01189834274754399308534160477
