| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 11-s − 13-s + 16-s + 18-s + 19-s − 22-s + 23-s − 26-s + 29-s − 31-s + 32-s + 36-s + 38-s − 43-s − 44-s + 46-s − 2·47-s + 49-s − 52-s + 58-s − 2·61-s − 62-s + 64-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 8-s + 9-s − 11-s − 13-s + 16-s + 18-s + 19-s − 22-s + 23-s − 26-s + 29-s − 31-s + 32-s + 36-s + 38-s − 43-s − 44-s + 46-s − 2·47-s + 49-s − 52-s + 58-s − 2·61-s − 62-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.303338364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.303338364\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−9.451090910271737869441272200118, −8.210346721278915778772859851097, −7.34067576540953108527614575660, −7.06453802486534580842693970504, −5.96171503832860490509432234987, −4.96963274052946164368422145510, −4.71449100694708243412791997382, −3.43904259109704609897937225722, −2.68787756596712922040962031221, −1.52917852974905540412251339942,
1.52917852974905540412251339942, 2.68787756596712922040962031221, 3.43904259109704609897937225722, 4.71449100694708243412791997382, 4.96963274052946164368422145510, 5.96171503832860490509432234987, 7.06453802486534580842693970504, 7.34067576540953108527614575660, 8.210346721278915778772859851097, 9.451090910271737869441272200118
