| L(s) = 1 | + 2-s + 4-s + 8-s + 2·9-s − 2·11-s + 16-s + 2·18-s − 2·22-s − 6·25-s + 32-s + 2·36-s − 24·43-s − 2·44-s − 7·49-s − 6·50-s + 64-s − 24·67-s + 2·72-s − 5·81-s − 24·86-s − 2·88-s − 7·98-s − 4·99-s − 6·100-s + 24·107-s + 20·113-s + 3·121-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 2/3·9-s − 0.603·11-s + 1/4·16-s + 0.471·18-s − 0.426·22-s − 6/5·25-s + 0.176·32-s + 1/3·36-s − 3.65·43-s − 0.301·44-s − 49-s − 0.848·50-s + 1/8·64-s − 2.93·67-s + 0.235·72-s − 5/9·81-s − 2.58·86-s − 0.213·88-s − 0.707·98-s − 0.402·99-s − 3/5·100-s + 2.32·107-s + 1.88·113-s + 3/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 758912 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 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
−7.956688321669486949443993345122, −7.44503492091689427214437925251, −7.28384383496805176593505108744, −6.58795954861217297502080464861, −6.22829773044480341331555491835, −5.83925184008878762369803223001, −5.11224655744256040433863219406, −4.88550380629058443410670715554, −4.39507406197561839253841275171, −3.72425550395436005402207348894, −3.31879301455778368319659817430, −2.74561130040773494403748773066, −1.89204299677024356271142659476, −1.50252589523985056842616413160, 0,
1.50252589523985056842616413160, 1.89204299677024356271142659476, 2.74561130040773494403748773066, 3.31879301455778368319659817430, 3.72425550395436005402207348894, 4.39507406197561839253841275171, 4.88550380629058443410670715554, 5.11224655744256040433863219406, 5.83925184008878762369803223001, 6.22829773044480341331555491835, 6.58795954861217297502080464861, 7.28384383496805176593505108744, 7.44503492091689427214437925251, 7.956688321669486949443993345122
