| L(s) = 1 | + 4-s + 9-s − 11-s − 13-s + 16-s − 19-s + 25-s − 29-s + 36-s − 44-s − 47-s + 49-s − 52-s − 53-s + 64-s − 67-s − 73-s − 76-s + 81-s − 97-s − 99-s + 100-s − 113-s − 116-s − 117-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 11-s − 13-s + 16-s − 19-s + 25-s − 29-s + 36-s − 44-s − 47-s + 49-s − 52-s − 53-s + 64-s − 67-s − 73-s − 76-s + 81-s − 97-s − 99-s + 100-s − 113-s − 116-s − 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075815530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.075815530\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−10.79713186765847637422532878004, −10.37433976262543325944254314351, −9.424206888504238465069665559576, −8.111038336031161999952469578703, −7.34449400323765820528258060318, −6.66401339390559653425935296281, −5.48690654260198598586905443130, −4.42456534739369053074658517116, −2.95828819358308552118150928765, −1.88532332578991924457925250353,
1.88532332578991924457925250353, 2.95828819358308552118150928765, 4.42456534739369053074658517116, 5.48690654260198598586905443130, 6.66401339390559653425935296281, 7.34449400323765820528258060318, 8.111038336031161999952469578703, 9.424206888504238465069665559576, 10.37433976262543325944254314351, 10.79713186765847637422532878004
