| L(s) = 1 | + 2.28·2-s + 3.15·3-s + 3.22·4-s + 2.12·5-s + 7.19·6-s + 2.78·8-s + 6.92·9-s + 4.85·10-s + 0.308·11-s + 10.1·12-s + 6.69·15-s − 0.0699·16-s − 1.77·17-s + 15.8·18-s + 1.78·19-s + 6.84·20-s + 0.704·22-s + 1.15·23-s + 8.78·24-s − 0.484·25-s + 12.3·27-s + 2.01·29-s + 15.2·30-s − 4.60·31-s − 5.73·32-s + 0.971·33-s − 4.05·34-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.81·3-s + 1.61·4-s + 0.950·5-s + 2.93·6-s + 0.985·8-s + 2.30·9-s + 1.53·10-s + 0.0930·11-s + 2.92·12-s + 1.72·15-s − 0.0174·16-s − 0.430·17-s + 3.72·18-s + 0.408·19-s + 1.53·20-s + 0.150·22-s + 0.239·23-s + 1.79·24-s − 0.0968·25-s + 2.37·27-s + 0.373·29-s + 2.79·30-s − 0.827·31-s − 1.01·32-s + 0.169·33-s − 0.695·34-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
| \(L(1)\) |
\(\approx\) |
\(12.98281161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.98281161\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.28T + 2T^{2} \) |
| 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 11 | \( 1 - 0.308T + 11T^{2} \) |
| 17 | \( 1 + 1.77T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 23 | \( 1 - 1.15T + 23T^{2} \) |
| 29 | \( 1 - 2.01T + 29T^{2} \) |
| 31 | \( 1 + 4.60T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 + 6.72T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 - 9.51T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 + 8.12T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 1.35T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 1.65T + 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−7.58918554004759190840454090875, −7.04619045586918501155690539068, −6.38928120559387101532398298456, −5.54206430890054363753209566968, −4.90946327853202426492180253911, −4.03715854876803341965629056269, −3.55699989956730669091011069959, −2.77063917324081097163121022899, −2.22878379984349794227960915567, −1.53261039030705094160793354414,
1.53261039030705094160793354414, 2.22878379984349794227960915567, 2.77063917324081097163121022899, 3.55699989956730669091011069959, 4.03715854876803341965629056269, 4.90946327853202426492180253911, 5.54206430890054363753209566968, 6.38928120559387101532398298456, 7.04619045586918501155690539068, 7.58918554004759190840454090875
