| L(s) = 1 | − 2-s + 4-s + 6·7-s − 8-s − 3·9-s − 6·14-s + 16-s + 3·18-s + 2·19-s − 6·25-s + 6·28-s − 14·29-s − 32-s − 3·36-s − 2·38-s + 12·41-s + 12·43-s + 14·49-s + 6·50-s + 6·53-s − 6·56-s + 14·58-s − 8·59-s − 12·61-s − 18·63-s + 64-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 2.26·7-s − 0.353·8-s − 9-s − 1.60·14-s + 1/4·16-s + 0.707·18-s + 0.458·19-s − 6/5·25-s + 1.13·28-s − 2.59·29-s − 0.176·32-s − 1/2·36-s − 0.324·38-s + 1.87·41-s + 1.82·43-s + 2·49-s + 0.848·50-s + 0.824·53-s − 0.801·56-s + 1.83·58-s − 1.04·59-s − 1.53·61-s − 2.26·63-s + 1/8·64-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.615764143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.615764143\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.199383346465115732806534764943, −7.79530737198181736366088991232, −7.53765502344261837304567389971, −7.27400754458840652751579174431, −6.39219962130712361368349883555, −5.81314493357481827808221159376, −5.60622467480311947759068539876, −5.19568868057375909855598413623, −4.51087827239310211962031199503, −4.07230082092463173625316334250, −3.46563852445574571937657654238, −2.63702473380429302423338873096, −2.03480291789933112683345231467, −1.67427855422752882506233657817, −0.68877860550954893610381081385,
0.68877860550954893610381081385, 1.67427855422752882506233657817, 2.03480291789933112683345231467, 2.63702473380429302423338873096, 3.46563852445574571937657654238, 4.07230082092463173625316334250, 4.51087827239310211962031199503, 5.19568868057375909855598413623, 5.60622467480311947759068539876, 5.81314493357481827808221159376, 6.39219962130712361368349883555, 7.27400754458840652751579174431, 7.53765502344261837304567389971, 7.79530737198181736366088991232, 8.199383346465115732806534764943
