| L(s) = 1 | − 0.852·2-s + 3-s − 1.27·4-s − 0.278·5-s − 0.852·6-s + 2.70·7-s + 2.79·8-s + 9-s + 0.237·10-s + 5.07·11-s − 1.27·12-s − 0.519·13-s − 2.30·14-s − 0.278·15-s + 0.166·16-s + 2.19·17-s − 0.852·18-s − 0.104·19-s + 0.354·20-s + 2.70·21-s − 4.32·22-s + 23-s + 2.79·24-s − 4.92·25-s + 0.443·26-s + 27-s − 3.44·28-s + ⋯ |
| L(s) = 1 | − 0.602·2-s + 0.577·3-s − 0.636·4-s − 0.124·5-s − 0.348·6-s + 1.02·7-s + 0.986·8-s + 0.333·9-s + 0.0750·10-s + 1.52·11-s − 0.367·12-s − 0.144·13-s − 0.616·14-s − 0.0718·15-s + 0.0415·16-s + 0.532·17-s − 0.200·18-s − 0.0239·19-s + 0.0791·20-s + 0.590·21-s − 0.921·22-s + 0.208·23-s + 0.569·24-s − 0.984·25-s + 0.0869·26-s + 0.192·27-s − 0.650·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.691750892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.691750892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.852T + 2T^{2} \) |
| 5 | \( 1 + 0.278T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 + 0.519T + 13T^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 + 0.104T + 19T^{2} \) |
| 31 | \( 1 - 7.35T + 31T^{2} \) |
| 37 | \( 1 + 0.259T + 37T^{2} \) |
| 41 | \( 1 - 0.509T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 1.04T + 53T^{2} \) |
| 59 | \( 1 - 0.323T + 59T^{2} \) |
| 61 | \( 1 - 8.63T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 0.479T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 + 6.67T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 - 0.587T + 97T^{2} \) |
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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
−9.082138870464007528909105016304, −8.400637765114990262355353795918, −7.895579505771250793197229115900, −7.11028562818665730746426499184, −6.05469639285091288458260454418, −4.86064022015876522195478495486, −4.27896662846464651262856595823, −3.40198460292905111669768662405, −1.88361616150578850373202807044, −1.02777379449661887524363401483,
1.02777379449661887524363401483, 1.88361616150578850373202807044, 3.40198460292905111669768662405, 4.27896662846464651262856595823, 4.86064022015876522195478495486, 6.05469639285091288458260454418, 7.11028562818665730746426499184, 7.895579505771250793197229115900, 8.400637765114990262355353795918, 9.082138870464007528909105016304
