| L(s) = 1 | + 3-s − 4-s − 7-s + 9-s − 12-s + 11·13-s − 3·16-s + 4·19-s − 21-s − 4·25-s + 27-s + 28-s + 16·31-s − 36-s + 5·37-s + 11·39-s + 6·43-s − 3·48-s − 6·49-s − 11·52-s + 4·57-s + 61-s − 63-s + 7·64-s − 67-s − 8·73-s − 4·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 0.377·7-s + 1/3·9-s − 0.288·12-s + 3.05·13-s − 3/4·16-s + 0.917·19-s − 0.218·21-s − 4/5·25-s + 0.192·27-s + 0.188·28-s + 2.87·31-s − 1/6·36-s + 0.821·37-s + 1.76·39-s + 0.914·43-s − 0.433·48-s − 6/7·49-s − 1.52·52-s + 0.529·57-s + 0.128·61-s − 0.125·63-s + 7/8·64-s − 0.122·67-s − 0.936·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450387 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450387 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431865396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431865396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 2383 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 18 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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
−8.494519726513593970683588230802, −8.251260915688354258093348840877, −7.919334175659462419432364489275, −7.23410865592357401371163084825, −6.59261070236340326493505402752, −6.30914115773479657824991870063, −5.89574365329184259884410619008, −5.35656411119314195767157210771, −4.46289985784418800532964052729, −4.23562058523972580108622454409, −3.69539915639052033331141253615, −3.09822301666632246132788287451, −2.61981136335623035604106011035, −1.51279474649104661782329063411, −0.923984668147151383047533714142,
0.923984668147151383047533714142, 1.51279474649104661782329063411, 2.61981136335623035604106011035, 3.09822301666632246132788287451, 3.69539915639052033331141253615, 4.23562058523972580108622454409, 4.46289985784418800532964052729, 5.35656411119314195767157210771, 5.89574365329184259884410619008, 6.30914115773479657824991870063, 6.59261070236340326493505402752, 7.23410865592357401371163084825, 7.919334175659462419432364489275, 8.251260915688354258093348840877, 8.494519726513593970683588230802
