| L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s − 2·11-s − 2·13-s − 4·15-s + 6·17-s + 4·21-s + 2·23-s − 7·25-s − 4·27-s + 10·29-s + 6·31-s + 4·33-s − 4·35-s − 14·37-s + 4·39-s + 4·41-s + 2·43-s + 6·45-s + 4·47-s + 3·49-s − 12·51-s − 2·53-s − 4·55-s + 4·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s − 0.603·11-s − 0.554·13-s − 1.03·15-s + 1.45·17-s + 0.872·21-s + 0.417·23-s − 7/5·25-s − 0.769·27-s + 1.85·29-s + 1.07·31-s + 0.696·33-s − 0.676·35-s − 2.30·37-s + 0.640·39-s + 0.624·41-s + 0.304·43-s + 0.894·45-s + 0.583·47-s + 3/7·49-s − 1.68·51-s − 0.274·53-s − 0.539·55-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13660416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13660416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.027473942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027473942\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 24 T + 273 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 207 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 242 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 158 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 6 T + 158 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.617605161969710662029179563299, −8.345074731644803461518263267674, −7.888777950753163610755844815408, −7.61196605610555987707552452838, −6.98870766625504171095633075218, −6.89419825735184798991796033444, −6.31164502516292163960713172810, −6.21181566003886440793031680112, −5.57474979918229926290593178050, −5.49986878442501105136738900869, −4.97525234525818746706608198425, −4.85256740027189898414080999623, −4.05590479552474677938598726157, −3.76546895843455998970723201874, −3.14493531295592046753827661780, −2.80085157086833620936431133335, −2.10236892868121322357607742976, −1.83170257304283884160587878345, −0.814068508232373021254646778160, −0.63052023412358617484836118400,
0.63052023412358617484836118400, 0.814068508232373021254646778160, 1.83170257304283884160587878345, 2.10236892868121322357607742976, 2.80085157086833620936431133335, 3.14493531295592046753827661780, 3.76546895843455998970723201874, 4.05590479552474677938598726157, 4.85256740027189898414080999623, 4.97525234525818746706608198425, 5.49986878442501105136738900869, 5.57474979918229926290593178050, 6.21181566003886440793031680112, 6.31164502516292163960713172810, 6.89419825735184798991796033444, 6.98870766625504171095633075218, 7.61196605610555987707552452838, 7.888777950753163610755844815408, 8.345074731644803461518263267674, 8.617605161969710662029179563299
