| L(s) = 1 | + 2-s − 4-s − 3·8-s + 6·9-s + 4·13-s − 16-s + 17-s + 6·18-s − 6·25-s + 4·26-s + 5·32-s + 34-s − 6·36-s − 2·49-s − 6·50-s − 4·52-s + 12·53-s + 7·64-s − 68-s − 18·72-s + 27·81-s − 20·89-s − 2·98-s + 6·100-s + 20·101-s − 12·104-s + 12·106-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 2·9-s + 1.10·13-s − 1/4·16-s + 0.242·17-s + 1.41·18-s − 6/5·25-s + 0.784·26-s + 0.883·32-s + 0.171·34-s − 36-s − 2/7·49-s − 0.848·50-s − 0.554·52-s + 1.64·53-s + 7/8·64-s − 0.121·68-s − 2.12·72-s + 3·81-s − 2.11·89-s − 0.202·98-s + 3/5·100-s + 1.99·101-s − 1.17·104-s + 1.16·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.060523199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.060523199\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 17 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−9.879462874821836240108491469219, −9.347932798696599941724487711285, −8.724067184873761626640889604437, −8.327057868773687890635908279232, −7.63295402223100965286156650330, −7.20329755564811241430390987512, −6.58378588415485042218422168420, −6.02601836072873175477515484739, −5.54032115876094107120502028665, −4.80794657601278007828941541338, −4.26542781169469736604809983472, −3.85300829032587940303250370471, −3.32267225710519999264467617420, −2.12416628352518724156461042862, −1.13084284529693092531563188617,
1.13084284529693092531563188617, 2.12416628352518724156461042862, 3.32267225710519999264467617420, 3.85300829032587940303250370471, 4.26542781169469736604809983472, 4.80794657601278007828941541338, 5.54032115876094107120502028665, 6.02601836072873175477515484739, 6.58378588415485042218422168420, 7.20329755564811241430390987512, 7.63295402223100965286156650330, 8.327057868773687890635908279232, 8.724067184873761626640889604437, 9.347932798696599941724487711285, 9.879462874821836240108491469219
