| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 2·13-s + 16-s + 2·17-s + 18-s − 5·25-s − 2·26-s − 8·29-s + 32-s + 2·34-s + 36-s + 10·37-s − 8·41-s + 6·49-s − 5·50-s − 2·52-s − 18·53-s − 8·58-s − 8·61-s + 64-s + 2·68-s + 72-s − 14·73-s + 10·74-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1/3·9-s − 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 25-s − 0.392·26-s − 1.48·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 1.64·37-s − 1.24·41-s + 6/7·49-s − 0.707·50-s − 0.277·52-s − 2.47·53-s − 1.05·58-s − 1.02·61-s + 1/8·64-s + 0.242·68-s + 0.117·72-s − 1.63·73-s + 1.16·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + p T^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 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
−7.77510822983141544964502159512, −7.37748159174788016825013627317, −6.96148688492685146251791510920, −6.33589861823542549196112231377, −6.03546905460441668600671064374, −5.58653164551121811859845627778, −5.10265876218290948980318728884, −4.61218896983376677193747038476, −4.18134746023244618069185112199, −3.67239845753938184692964394428, −3.14655710350619592913776277128, −2.59021833480832384888300657502, −1.88938600733915168858431639545, −1.34652919276860947300075499704, 0,
1.34652919276860947300075499704, 1.88938600733915168858431639545, 2.59021833480832384888300657502, 3.14655710350619592913776277128, 3.67239845753938184692964394428, 4.18134746023244618069185112199, 4.61218896983376677193747038476, 5.10265876218290948980318728884, 5.58653164551121811859845627778, 6.03546905460441668600671064374, 6.33589861823542549196112231377, 6.96148688492685146251791510920, 7.37748159174788016825013627317, 7.77510822983141544964502159512
