| 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 − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-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 − 49-s − 50-s − 2·52-s + 2·53-s + 64-s + 68-s + 72-s + 81-s − 2·89-s + 98-s + 100-s − 2·101-s + 2·104-s − 2·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3327885867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3327885867\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T^{2} \) |
| 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
−15.02449061344557603691883920300, −14.36118879225983677422230504881, −12.47090892005841483622787798176, −11.65059393536294409108661621379, −10.36498251692829322753123217545, −9.378387468381204143197364963903, −8.141436839216771787296709142564, −7.01903598371148299622349237377, −5.41265195424507937916978031210, −2.75562279580887633473300699062,
2.75562279580887633473300699062, 5.41265195424507937916978031210, 7.01903598371148299622349237377, 8.141436839216771787296709142564, 9.378387468381204143197364963903, 10.36498251692829322753123217545, 11.65059393536294409108661621379, 12.47090892005841483622787798176, 14.36118879225983677422230504881, 15.02449061344557603691883920300
