| L(s) = 1 | + 3·2-s − 3-s + 4·4-s − 3·6-s + 3·8-s + 3·9-s + 9·11-s − 4·12-s + 2·13-s + 3·16-s + 6·17-s + 9·18-s − 3·19-s + 27·22-s − 3·24-s + 7·25-s + 6·26-s − 8·27-s − 3·29-s + 6·32-s − 9·33-s + 18·34-s + 12·36-s − 9·38-s − 2·39-s + 9·41-s − 11·43-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 2·4-s − 1.22·6-s + 1.06·8-s + 9-s + 2.71·11-s − 1.15·12-s + 0.554·13-s + 3/4·16-s + 1.45·17-s + 2.12·18-s − 0.688·19-s + 5.75·22-s − 0.612·24-s + 7/5·25-s + 1.17·26-s − 1.53·27-s − 0.557·29-s + 1.06·32-s − 1.56·33-s + 3.08·34-s + 2·36-s − 1.45·38-s − 0.320·39-s + 1.40·41-s − 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.383879466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.383879466\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T + 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 68 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 71 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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
−11.06663851358317886124815166548, −10.54063427834550937794503280637, −9.955539897883459469536510071681, −9.666191730033022348124237170005, −8.950557914232390099311250175246, −8.895299466189142893195785011950, −7.905187421202558064518566865714, −7.57312751345662969479178831677, −6.87717086522113280013457682095, −6.55374477248910704439931241286, −6.09104426320387020544838645587, −5.88005622856155511040273888957, −5.17276796593610130225995289728, −4.69506539046643790491797726568, −4.31436506008169208408382198876, −3.72953607506592133535507672215, −3.64388388385592664360945142319, −2.91454216147683498369741216950, −1.44708519777574172275652910078, −1.40760246408540824846251294542,
1.40760246408540824846251294542, 1.44708519777574172275652910078, 2.91454216147683498369741216950, 3.64388388385592664360945142319, 3.72953607506592133535507672215, 4.31436506008169208408382198876, 4.69506539046643790491797726568, 5.17276796593610130225995289728, 5.88005622856155511040273888957, 6.09104426320387020544838645587, 6.55374477248910704439931241286, 6.87717086522113280013457682095, 7.57312751345662969479178831677, 7.905187421202558064518566865714, 8.895299466189142893195785011950, 8.950557914232390099311250175246, 9.666191730033022348124237170005, 9.955539897883459469536510071681, 10.54063427834550937794503280637, 11.06663851358317886124815166548
