| L(s) = 1 | − 1.56·2-s − 2.56·3-s + 0.438·4-s + 4·6-s + 2.43·8-s + 3.56·9-s + 2.56·11-s − 1.12·12-s + 4.56·13-s − 4.68·16-s − 4.56·17-s − 5.56·18-s − 1.12·19-s − 4·22-s + 5.12·23-s − 6.24·24-s − 7.12·26-s − 1.43·27-s − 5.68·29-s + 2.43·32-s − 6.56·33-s + 7.12·34-s + 1.56·36-s − 6·37-s + 1.75·38-s − 11.6·39-s + 3.12·41-s + ⋯ |
| L(s) = 1 | − 1.10·2-s − 1.47·3-s + 0.219·4-s + 1.63·6-s + 0.862·8-s + 1.18·9-s + 0.772·11-s − 0.324·12-s + 1.26·13-s − 1.17·16-s − 1.10·17-s − 1.31·18-s − 0.257·19-s − 0.852·22-s + 1.06·23-s − 1.27·24-s − 1.39·26-s − 0.276·27-s − 1.05·29-s + 0.431·32-s − 1.14·33-s + 1.22·34-s + 0.260·36-s − 0.986·37-s + 0.284·38-s − 1.87·39-s + 0.487·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4563986464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4563986464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 9.36T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.733638090666761017259539380068, −8.913493558180477633687544787729, −8.363945755426394392214804927125, −7.03572311954138102241617287248, −6.62905558030012387741304124352, −5.62968655311004813749303885056, −4.71477142386238743600083937442, −3.77905304803625332849037981240, −1.73737059387617932385186526788, −0.66777470470393653869777273786,
0.66777470470393653869777273786, 1.73737059387617932385186526788, 3.77905304803625332849037981240, 4.71477142386238743600083937442, 5.62968655311004813749303885056, 6.62905558030012387741304124352, 7.03572311954138102241617287248, 8.363945755426394392214804927125, 8.913493558180477633687544787729, 9.733638090666761017259539380068
