| L(s) = 1 | − 2·3-s + 2·5-s + 2·7-s + 2·9-s − 4·11-s + 6·13-s − 4·15-s + 2·19-s − 4·21-s + 8·23-s − 2·25-s − 6·27-s + 4·31-s + 8·33-s + 4·35-s − 12·39-s − 8·41-s + 4·43-s + 4·45-s + 12·47-s + 3·49-s − 20·53-s − 8·55-s − 4·57-s − 14·59-s + 18·61-s + 4·63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 0.755·7-s + 2/3·9-s − 1.20·11-s + 1.66·13-s − 1.03·15-s + 0.458·19-s − 0.872·21-s + 1.66·23-s − 2/5·25-s − 1.15·27-s + 0.718·31-s + 1.39·33-s + 0.676·35-s − 1.92·39-s − 1.24·41-s + 0.609·43-s + 0.596·45-s + 1.75·47-s + 3/7·49-s − 2.74·53-s − 1.07·55-s − 0.529·57-s − 1.82·59-s + 2.30·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.198544877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.198544877\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−12.57147821511774555112762475164, −11.89128870713481800423031329694, −11.34032462353600206896224182854, −11.01288969708337270018928307882, −10.80313266572489459987418726777, −10.26540356301178488203703941407, −9.540661422323866214856672503338, −9.320133094213933446818406373524, −8.368495865601782899520106969985, −8.172233611398313880280940954613, −7.43086658449387775965214269264, −6.81721882641465152602144438016, −6.16847522482240139012874241475, −5.78848950550294983794700031528, −5.20401025158157507787190827967, −4.93937829414410636222363652171, −3.97854214530522441719437038940, −3.10997951639284746193994414008, −2.06607972394989705631620508459, −1.09406892649004172180256059289,
1.09406892649004172180256059289, 2.06607972394989705631620508459, 3.10997951639284746193994414008, 3.97854214530522441719437038940, 4.93937829414410636222363652171, 5.20401025158157507787190827967, 5.78848950550294983794700031528, 6.16847522482240139012874241475, 6.81721882641465152602144438016, 7.43086658449387775965214269264, 8.172233611398313880280940954613, 8.368495865601782899520106969985, 9.320133094213933446818406373524, 9.540661422323866214856672503338, 10.26540356301178488203703941407, 10.80313266572489459987418726777, 11.01288969708337270018928307882, 11.34032462353600206896224182854, 11.89128870713481800423031329694, 12.57147821511774555112762475164
