| L(s) = 1 | − 2-s + 4-s − 8-s − 9-s + 2·13-s + 16-s − 17-s + 18-s + 25-s − 2·26-s − 32-s + 34-s − 36-s − 50-s + 2·52-s + 2·53-s + 64-s − 68-s + 72-s + 81-s + 2·89-s + 100-s + 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s − 9-s + 2·13-s + 16-s − 17-s + 18-s + 25-s − 2·26-s − 32-s + 34-s − 36-s − 50-s + 2·52-s + 2·53-s + 64-s − 68-s + 72-s + 81-s + 2·89-s + 100-s + 2·101-s − 2·104-s − 2·106-s − 2·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7942650089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7942650089\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.734033661653060851565056647550, −8.398598074443980036737263971839, −7.45073921304838053001412056658, −6.51371192282355604081864635128, −6.12020281721816720237542000126, −5.22844880036876663089043014736, −3.90737991840690673651127826714, −3.08420048273440997718033667649, −2.12851263876107765259720395327, −0.922737046567251908763524139817,
0.922737046567251908763524139817, 2.12851263876107765259720395327, 3.08420048273440997718033667649, 3.90737991840690673651127826714, 5.22844880036876663089043014736, 6.12020281721816720237542000126, 6.51371192282355604081864635128, 7.45073921304838053001412056658, 8.398598074443980036737263971839, 8.734033661653060851565056647550
