| L(s) = 1 | − 0.174·2-s + 2.31·3-s − 1.96·4-s + 1.01·5-s − 0.405·6-s + 0.693·8-s + 2.37·9-s − 0.177·10-s − 4.96·11-s − 4.56·12-s + 5.62·13-s + 2.35·15-s + 3.81·16-s − 3.76·17-s − 0.415·18-s − 3.14·19-s − 2.00·20-s + 0.868·22-s + 4.19·23-s + 1.60·24-s − 3.96·25-s − 0.982·26-s − 1.44·27-s − 7.96·29-s − 0.412·30-s − 9.91·31-s − 2.05·32-s + ⋯ |
| L(s) = 1 | − 0.123·2-s + 1.33·3-s − 0.984·4-s + 0.454·5-s − 0.165·6-s + 0.245·8-s + 0.791·9-s − 0.0562·10-s − 1.49·11-s − 1.31·12-s + 1.55·13-s + 0.608·15-s + 0.954·16-s − 0.913·17-s − 0.0978·18-s − 0.720·19-s − 0.447·20-s + 0.185·22-s + 0.874·23-s + 0.328·24-s − 0.793·25-s − 0.192·26-s − 0.278·27-s − 1.47·29-s − 0.0752·30-s − 1.78·31-s − 0.363·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.174T + 2T^{2} \) |
| 3 | \( 1 - 2.31T + 3T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 11 | \( 1 + 4.96T + 11T^{2} \) |
| 13 | \( 1 - 5.62T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 + 3.14T + 19T^{2} \) |
| 23 | \( 1 - 4.19T + 23T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 + 9.91T + 31T^{2} \) |
| 37 | \( 1 + 0.106T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 - 0.0223T + 43T^{2} \) |
| 47 | \( 1 + 5.74T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 8.90T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 + 9.29T + 89T^{2} \) |
| 97 | \( 1 - 4.37T + 97T^{2} \) |
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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
−8.452174230152540825341999068933, −7.70078370458092110465717826951, −6.81473549529262749572727631478, −5.63555868091334088192011524789, −5.19622751601370339061620264951, −3.91376451340783790419875958698, −3.59784599214468311319449063491, −2.45167386927747753276206620739, −1.67925901702707451122548160848, 0,
1.67925901702707451122548160848, 2.45167386927747753276206620739, 3.59784599214468311319449063491, 3.91376451340783790419875958698, 5.19622751601370339061620264951, 5.63555868091334088192011524789, 6.81473549529262749572727631478, 7.70078370458092110465717826951, 8.452174230152540825341999068933
