| L(s) = 1 | + 0.470·2-s − 2.24·3-s − 1.77·4-s − 0.529·5-s − 1.05·6-s − 1.77·8-s + 2.05·9-s − 0.249·10-s − 2.24·11-s + 4.00·12-s − 13-s + 1.19·15-s + 2.71·16-s + 1.30·17-s + 0.968·18-s + 1.47·19-s + 0.941·20-s − 1.05·22-s + 5.83·23-s + 4.00·24-s − 4.71·25-s − 0.470·26-s + 2.11·27-s + 5.22·29-s + 0.560·30-s + 7.02·31-s + 4.83·32-s + ⋯ |
| L(s) = 1 | + 0.332·2-s − 1.29·3-s − 0.889·4-s − 0.236·5-s − 0.432·6-s − 0.628·8-s + 0.686·9-s − 0.0787·10-s − 0.678·11-s + 1.15·12-s − 0.277·13-s + 0.307·15-s + 0.679·16-s + 0.317·17-s + 0.228·18-s + 0.337·19-s + 0.210·20-s − 0.225·22-s + 1.21·23-s + 0.816·24-s − 0.943·25-s − 0.0923·26-s + 0.407·27-s + 0.969·29-s + 0.102·30-s + 1.26·31-s + 0.855·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6687979106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6687979106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.470T + 2T^{2} \) |
| 3 | \( 1 + 2.24T + 3T^{2} \) |
| 5 | \( 1 + 0.529T + 5T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 1.47T + 19T^{2} \) |
| 23 | \( 1 - 5.83T + 23T^{2} \) |
| 29 | \( 1 - 5.22T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + 7.64T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 6.91T + 89T^{2} \) |
| 97 | \( 1 - 3.47T + 97T^{2} \) |
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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
−10.53479542699030671970451914933, −9.983697240012248097291284913560, −8.858273253139612343567181957335, −7.936017168618540638475479885292, −6.81802789260794889648396775126, −5.76250169193704556090730717133, −5.13183137493496915966746099193, −4.36088514355770213159537729886, −3.00857864923386076986054653525, −0.71272129291482955258320604293,
0.71272129291482955258320604293, 3.00857864923386076986054653525, 4.36088514355770213159537729886, 5.13183137493496915966746099193, 5.76250169193704556090730717133, 6.81802789260794889648396775126, 7.936017168618540638475479885292, 8.858273253139612343567181957335, 9.983697240012248097291284913560, 10.53479542699030671970451914933
