| L(s) = 1 | + 2.44·2-s + 3-s + 4.00·4-s − 0.790·5-s + 2.44·6-s + 7-s + 4.90·8-s + 9-s − 1.93·10-s + 4.91·11-s + 4.00·12-s + 5.52·13-s + 2.44·14-s − 0.790·15-s + 4.00·16-s + 2.59·17-s + 2.44·18-s − 3.10·19-s − 3.16·20-s + 21-s + 12.0·22-s − 6.73·23-s + 4.90·24-s − 4.37·25-s + 13.5·26-s + 27-s + 4.00·28-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 0.577·3-s + 2.00·4-s − 0.353·5-s + 1.00·6-s + 0.377·7-s + 1.73·8-s + 0.333·9-s − 0.612·10-s + 1.48·11-s + 1.15·12-s + 1.53·13-s + 0.654·14-s − 0.204·15-s + 1.00·16-s + 0.629·17-s + 0.577·18-s − 0.712·19-s − 0.707·20-s + 0.218·21-s + 2.56·22-s − 1.40·23-s + 1.00·24-s − 0.874·25-s + 2.65·26-s + 0.192·27-s + 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.671410832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.671410832\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 - T \) |
| good | 2 | \( 1 - 2.44T + 2T^{2} \) |
| 5 | \( 1 + 0.790T + 5T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 - 5.52T + 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 + 6.73T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 31 | \( 1 + 5.20T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2.55T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 8.42T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 - 8.92T + 59T^{2} \) |
| 61 | \( 1 - 5.13T + 61T^{2} \) |
| 67 | \( 1 - 0.982T + 67T^{2} \) |
| 71 | \( 1 - 4.00T + 71T^{2} \) |
| 73 | \( 1 - 5.74T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 1.35T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.322221273334496069949712968555, −7.56574613827795758424751026728, −6.69665382664380985557031629570, −6.08655178527632049047062592092, −5.50131084372879172794513697515, −4.23238639381808682321108969661, −3.95273728504081201068869895855, −3.45163742276073242976753122518, −2.21184575943601929274382785931, −1.41183654447234987261695170698,
1.41183654447234987261695170698, 2.21184575943601929274382785931, 3.45163742276073242976753122518, 3.95273728504081201068869895855, 4.23238639381808682321108969661, 5.50131084372879172794513697515, 6.08655178527632049047062592092, 6.69665382664380985557031629570, 7.56574613827795758424751026728, 8.322221273334496069949712968555
