L(s) = 1 | − 2·3-s − 3·4-s + 3·9-s + 6·12-s + 5·16-s − 14·17-s − 12·23-s − 9·25-s − 4·27-s − 2·29-s − 9·36-s + 12·43-s − 10·48-s − 10·49-s + 28·51-s − 18·53-s + 2·61-s − 3·64-s + 42·68-s + 24·69-s + 18·75-s − 8·79-s + 5·81-s + 4·87-s + 36·92-s + 27·100-s + 6·101-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 3/2·4-s + 9-s + 1.73·12-s + 5/4·16-s − 3.39·17-s − 2.50·23-s − 9/5·25-s − 0.769·27-s − 0.371·29-s − 3/2·36-s + 1.82·43-s − 1.44·48-s − 1.42·49-s + 3.92·51-s − 2.47·53-s + 0.256·61-s − 3/8·64-s + 5.09·68-s + 2.88·69-s + 2.07·75-s − 0.900·79-s + 5/9·81-s + 0.428·87-s + 3.75·92-s + 2.69·100-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 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
−8.551927544281445958636300086280, −7.83924221365301722237117171435, −7.81599875388453611751040547561, −6.79243034337170992835505863348, −6.47571815039088450225769754472, −5.91564610472601612817558896864, −5.65964626382136303973235650945, −4.77533644988124531739406678713, −4.47621499216397670225811127054, −4.16029256597568730693149405009, −3.67795572665658491238814173509, −2.32792075963527314368769910890, −1.75898568134433752432491320831, 0, 0,
1.75898568134433752432491320831, 2.32792075963527314368769910890, 3.67795572665658491238814173509, 4.16029256597568730693149405009, 4.47621499216397670225811127054, 4.77533644988124531739406678713, 5.65964626382136303973235650945, 5.91564610472601612817558896864, 6.47571815039088450225769754472, 6.79243034337170992835505863348, 7.81599875388453611751040547561, 7.83924221365301722237117171435, 8.551927544281445958636300086280