| L(s) = 1 | + 1.55·2-s + 0.489·3-s + 0.417·4-s + 1.19·5-s + 0.760·6-s − 2.46·8-s − 2.76·9-s + 1.85·10-s + 2.11·11-s + 0.204·12-s + 0.582·15-s − 4.66·16-s + 0.906·17-s − 4.29·18-s − 6.69·19-s + 0.496·20-s + 3.28·22-s + 3.59·23-s − 1.20·24-s − 3.58·25-s − 2.81·27-s + 8.51·29-s + 0.906·30-s + 5.28·31-s − 2.32·32-s + 1.03·33-s + 1.40·34-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.282·3-s + 0.208·4-s + 0.532·5-s + 0.310·6-s − 0.870·8-s − 0.920·9-s + 0.585·10-s + 0.638·11-s + 0.0589·12-s + 0.150·15-s − 1.16·16-s + 0.219·17-s − 1.01·18-s − 1.53·19-s + 0.111·20-s + 0.701·22-s + 0.750·23-s − 0.245·24-s − 0.716·25-s − 0.542·27-s + 1.58·29-s + 0.165·30-s + 0.949·31-s − 0.410·32-s + 0.180·33-s + 0.241·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.549423814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.549423814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 3 | \( 1 - 0.489T + 3T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 11 | \( 1 - 2.11T + 11T^{2} \) |
| 17 | \( 1 - 0.906T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 3.18T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 - 8.26T + 61T^{2} \) |
| 67 | \( 1 + 3.74T + 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 5.73T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + 6.20T + 97T^{2} \) |
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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
−7.910297933708158744892479080875, −6.65246738655942049887967630959, −6.34870980810783436849239907685, −5.69991337822895711697424512846, −4.96345998979798284807520924038, −4.27806630549629438351372499903, −3.60971163568122848774035920368, −2.72636566925567510569641266622, −2.21939882375244389866867841860, −0.75472326949701409360089523722,
0.75472326949701409360089523722, 2.21939882375244389866867841860, 2.72636566925567510569641266622, 3.60971163568122848774035920368, 4.27806630549629438351372499903, 4.96345998979798284807520924038, 5.69991337822895711697424512846, 6.34870980810783436849239907685, 6.65246738655942049887967630959, 7.910297933708158744892479080875
