| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 11-s + 12-s − 13-s + 16-s + 17-s + 22-s − 2·23-s + 24-s + 25-s − 26-s − 27-s − 2·31-s + 32-s + 33-s + 34-s − 39-s + 44-s − 2·46-s + 48-s + 50-s + 51-s − 52-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 11-s + 12-s − 13-s + 16-s + 17-s + 22-s − 2·23-s + 24-s + 25-s − 26-s − 27-s − 2·31-s + 32-s + 33-s + 34-s − 39-s + 44-s − 2·46-s + 48-s + 50-s + 51-s − 52-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.311204680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.311204680\) |
| \(L(1)\) |
|
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 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 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 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−8.750923436368834414603330497536, −7.83391507264668844340343737627, −7.43004858406289332920532983976, −6.48631318380755617525975688312, −5.72681217986316193188470379777, −4.92180405279032661637103741160, −3.88621534427499322473961829782, −3.44323487701105061188141418079, −2.45818357139276637353272527240, −1.67914071433243041950453158480,
1.67914071433243041950453158480, 2.45818357139276637353272527240, 3.44323487701105061188141418079, 3.88621534427499322473961829782, 4.92180405279032661637103741160, 5.72681217986316193188470379777, 6.48631318380755617525975688312, 7.43004858406289332920532983976, 7.83391507264668844340343737627, 8.750923436368834414603330497536
