| L(s) = 1 | − 2-s + 3-s − 6-s + 7-s + 8-s + 9-s + 11-s + 13-s − 14-s − 16-s − 17-s − 18-s + 21-s − 22-s + 24-s − 26-s + 27-s − 29-s + 33-s + 34-s + 39-s − 2·41-s − 42-s − 47-s − 48-s − 51-s − 54-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 6-s + 7-s + 8-s + 9-s + 11-s + 13-s − 14-s − 16-s − 17-s − 18-s + 21-s − 22-s + 24-s − 26-s + 27-s − 29-s + 33-s + 34-s + 39-s − 2·41-s − 42-s − 47-s − 48-s − 51-s − 54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.134814488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.134814488\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 + T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.153959867801180873893851027942, −8.398351881897981225837760183037, −8.197901742371325386932054234321, −7.18162872944690970488579657090, −6.51619990310614610270293874950, −5.07453381302967518027552296225, −4.25366982859792321760234730509, −3.53528095117272459216884786099, −1.97497682296563613442347329683, −1.37778745196513544815207440794,
1.37778745196513544815207440794, 1.97497682296563613442347329683, 3.53528095117272459216884786099, 4.25366982859792321760234730509, 5.07453381302967518027552296225, 6.51619990310614610270293874950, 7.18162872944690970488579657090, 8.197901742371325386932054234321, 8.398351881897981225837760183037, 9.153959867801180873893851027942
