| L(s) = 1 | + 4·3-s + 6·9-s + 6·11-s + 10·17-s + 2·19-s − 9·25-s − 4·27-s + 24·33-s + 12·41-s + 14·43-s − 5·49-s + 40·51-s + 8·57-s − 28·59-s − 30·73-s − 36·75-s − 37·81-s − 8·83-s + 32·97-s + 36·99-s − 20·107-s + 4·113-s + 5·121-s + 48·123-s + 127-s + 56·129-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·9-s + 1.80·11-s + 2.42·17-s + 0.458·19-s − 9/5·25-s − 0.769·27-s + 4.17·33-s + 1.87·41-s + 2.13·43-s − 5/7·49-s + 5.60·51-s + 1.05·57-s − 3.64·59-s − 3.51·73-s − 4.15·75-s − 4.11·81-s − 0.878·83-s + 3.24·97-s + 3.61·99-s − 1.93·107-s + 0.376·113-s + 5/11·121-s + 4.32·123-s + 0.0887·127-s + 4.93·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.804397963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.804397963\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
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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
−8.865784190712433894154140882392, −8.186562113914066026677579677805, −7.79236899027639545070642498110, −7.43537131861627444263317650332, −7.41110136347227768123475878009, −6.20879084669067007942487517486, −5.89904763388088518363904025264, −5.62405161804576690819108289841, −4.35808362404597816678481770906, −4.19084046620099313866498183608, −3.41145508666670040670114085640, −3.25598629803065603741482539939, −2.67622771685970545269516699451, −1.81226351589066908089801551987, −1.26674131115266326801058270565,
1.26674131115266326801058270565, 1.81226351589066908089801551987, 2.67622771685970545269516699451, 3.25598629803065603741482539939, 3.41145508666670040670114085640, 4.19084046620099313866498183608, 4.35808362404597816678481770906, 5.62405161804576690819108289841, 5.89904763388088518363904025264, 6.20879084669067007942487517486, 7.41110136347227768123475878009, 7.43537131861627444263317650332, 7.79236899027639545070642498110, 8.186562113914066026677579677805, 8.865784190712433894154140882392
