| L(s) = 1 | − 2-s − 0.303·3-s + 4-s + 0.0493·5-s + 0.303·6-s − 2.29·7-s − 8-s − 2.90·9-s − 0.0493·10-s − 4.83·11-s − 0.303·12-s − 5.48·13-s + 2.29·14-s − 0.0149·15-s + 16-s + 1.01·17-s + 2.90·18-s − 1.82·19-s + 0.0493·20-s + 0.695·21-s + 4.83·22-s − 23-s + 0.303·24-s − 4.99·25-s + 5.48·26-s + 1.79·27-s − 2.29·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.175·3-s + 0.5·4-s + 0.0220·5-s + 0.123·6-s − 0.866·7-s − 0.353·8-s − 0.969·9-s − 0.0155·10-s − 1.45·11-s − 0.0875·12-s − 1.51·13-s + 0.612·14-s − 0.00386·15-s + 0.250·16-s + 0.245·17-s + 0.685·18-s − 0.419·19-s + 0.0110·20-s + 0.151·21-s + 1.03·22-s − 0.208·23-s + 0.0619·24-s − 0.999·25-s + 1.07·26-s + 0.344·27-s − 0.433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0005606800013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0005606800013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.303T + 3T^{2} \) |
| 5 | \( 1 - 0.0493T + 5T^{2} \) |
| 7 | \( 1 + 2.29T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 + 5.48T + 13T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 + 1.82T + 19T^{2} \) |
| 29 | \( 1 - 1.97T + 29T^{2} \) |
| 31 | \( 1 + 9.42T + 31T^{2} \) |
| 37 | \( 1 - 2.10T + 37T^{2} \) |
| 41 | \( 1 + 8.83T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 4.15T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 7.54T + 73T^{2} \) |
| 79 | \( 1 - 0.850T + 79T^{2} \) |
| 83 | \( 1 + 5.36T + 83T^{2} \) |
| 89 | \( 1 - 5.39T + 89T^{2} \) |
| 97 | \( 1 + 0.459T + 97T^{2} \) |
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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
−7.990904035632947730397967812638, −7.50045852421812414127491975852, −6.82484061370703887690230080497, −5.85060640808767040089261002206, −5.48784320361214313979588899402, −4.55796610592818708793911434164, −3.28578398662507884627047017842, −2.72481329498000825296492654277, −1.93504535855671344052946443696, −0.01270695488059378935188962064,
0.01270695488059378935188962064, 1.93504535855671344052946443696, 2.72481329498000825296492654277, 3.28578398662507884627047017842, 4.55796610592818708793911434164, 5.48784320361214313979588899402, 5.85060640808767040089261002206, 6.82484061370703887690230080497, 7.50045852421812414127491975852, 7.990904035632947730397967812638
