| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s + 8·6-s + 4·7-s + 8·9-s − 8·12-s + 26·13-s − 8·14-s − 4·16-s + 34·17-s − 16·18-s + 16·19-s − 16·21-s + 36·23-s − 25·25-s − 52·26-s − 36·27-s + 8·28-s + 8·32-s − 68·34-s + 16·36-s − 86·37-s − 32·38-s − 104·39-s + 32·42-s − 4·43-s + ⋯ |
| L(s) = 1 | − 2-s − 4/3·3-s + 1/2·4-s + 4/3·6-s + 4/7·7-s + 8/9·9-s − 2/3·12-s + 2·13-s − 4/7·14-s − 1/4·16-s + 2·17-s − 8/9·18-s + 0.842·19-s − 0.761·21-s + 1.56·23-s − 25-s − 2·26-s − 4/3·27-s + 2/7·28-s + 1/4·32-s − 2·34-s + 4/9·36-s − 2.32·37-s − 0.842·38-s − 8/3·39-s + 0.761·42-s − 0.0930·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16900 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7857435170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7857435170\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 158 T^{2} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 86 T + 3698 T^{2} + 86 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3262 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 44 T + 968 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T + 338 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 112 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8482 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )( 1 + 110 T + p^{2} T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 1918 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 44 T + 968 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 66 T + 2178 T^{2} + 66 p^{2} T^{3} + p^{4} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.22887623555332381236373412829, −12.68143861608197412592345106143, −11.99455002783468358983178759317, −11.65737010875452002846007548506, −11.16484451060084725142688729329, −10.90292816643828378046288536594, −10.15014099255294551691877897257, −9.942842535234986450374803335510, −9.089915945262003620367873461091, −8.597679627697170880096830863508, −8.051301431153604804417727448238, −7.36931099472672890617723486896, −6.94336015610517485174488442102, −6.06321908999521924511629452443, −5.39647144297495123359387038652, −5.34362165452267136333468052486, −3.98230745789075088341109485919, −3.33452187059417402387053512670, −1.56919457465351752253050121712, −0.882615403518093266353684998160,
0.882615403518093266353684998160, 1.56919457465351752253050121712, 3.33452187059417402387053512670, 3.98230745789075088341109485919, 5.34362165452267136333468052486, 5.39647144297495123359387038652, 6.06321908999521924511629452443, 6.94336015610517485174488442102, 7.36931099472672890617723486896, 8.051301431153604804417727448238, 8.597679627697170880096830863508, 9.089915945262003620367873461091, 9.942842535234986450374803335510, 10.15014099255294551691877897257, 10.90292816643828378046288536594, 11.16484451060084725142688729329, 11.65737010875452002846007548506, 11.99455002783468358983178759317, 12.68143861608197412592345106143, 13.22887623555332381236373412829
