| L(s) = 1 | − 3·4-s + 9-s − 8·11-s − 4·13-s + 5·16-s + 4·17-s + 25-s − 4·29-s − 3·36-s − 20·37-s + 8·43-s + 24·44-s + 16·47-s − 14·49-s + 12·52-s − 10·53-s − 8·59-s − 3·64-s − 12·68-s + 81-s − 12·89-s + 4·97-s − 8·99-s − 3·100-s − 24·107-s + 4·113-s + 12·116-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1/3·9-s − 2.41·11-s − 1.10·13-s + 5/4·16-s + 0.970·17-s + 1/5·25-s − 0.742·29-s − 1/2·36-s − 3.28·37-s + 1.21·43-s + 3.61·44-s + 2.33·47-s − 2·49-s + 1.66·52-s − 1.37·53-s − 1.04·59-s − 3/8·64-s − 1.45·68-s + 1/9·81-s − 1.27·89-s + 0.406·97-s − 0.804·99-s − 0.299·100-s − 2.32·107-s + 0.376·113-s + 1.11·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 632025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 632025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2694586911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2694586911\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 53 | $C_2$ | \( 1 + 10 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.392917483718792305533375912226, −7.84064739199926515245088346140, −7.66488013441745380243230842523, −7.25044964749836606151779972659, −6.67099519053434096120034923120, −5.79877811946485867581107846427, −5.33353989962921287399929873808, −5.23920392624592057055772361749, −4.75080284173768807466029861290, −4.24354520384694878555714670338, −3.55071345479709815309520788186, −3.03661931965604314046655893326, −2.44016448217740808263005926859, −1.56365762830737101813907673272, −0.26299167498970711005220848982,
0.26299167498970711005220848982, 1.56365762830737101813907673272, 2.44016448217740808263005926859, 3.03661931965604314046655893326, 3.55071345479709815309520788186, 4.24354520384694878555714670338, 4.75080284173768807466029861290, 5.23920392624592057055772361749, 5.33353989962921287399929873808, 5.79877811946485867581107846427, 6.67099519053434096120034923120, 7.25044964749836606151779972659, 7.66488013441745380243230842523, 7.84064739199926515245088346140, 8.392917483718792305533375912226
