L(s) = 1 | − 2-s − 3-s + 4-s + 2.41·5-s + 6-s − 8-s + 9-s − 2.41·10-s + 2·11-s − 12-s − 3.82·13-s − 2.41·15-s + 16-s − 5.24·17-s − 18-s + 7.65·19-s + 2.41·20-s − 2·22-s + 23-s + 24-s + 0.828·25-s + 3.82·26-s − 27-s − 6.82·29-s + 2.41·30-s + 2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.07·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.763·10-s + 0.603·11-s − 0.288·12-s − 1.06·13-s − 0.623·15-s + 0.250·16-s − 1.27·17-s − 0.235·18-s + 1.75·19-s + 0.539·20-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 0.165·25-s + 0.750·26-s − 0.192·27-s − 1.26·29-s + 0.440·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2.41T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.82T + 13T^{2} \) |
| 17 | \( 1 + 5.24T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 5.17T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 - 4.07T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 4.89T + 67T^{2} \) |
| 71 | \( 1 + 4.17T + 71T^{2} \) |
| 73 | \( 1 + 0.656T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 7.17T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−7.35555364930301269473262971884, −7.06637445993770824016970542142, −6.29114639329103663611588379297, −5.54433240816924753327134644685, −5.06191074620854839737914845218, −4.02420443094422676565869295832, −2.90946007783184454038611896724, −2.03371527477243769959777579327, −1.30153165967718997868510119324, 0,
1.30153165967718997868510119324, 2.03371527477243769959777579327, 2.90946007783184454038611896724, 4.02420443094422676565869295832, 5.06191074620854839737914845218, 5.54433240816924753327134644685, 6.29114639329103663611588379297, 7.06637445993770824016970542142, 7.35555364930301269473262971884