| L(s) = 1 | − 3-s + 0.561·5-s + 2.56·7-s + 9-s − 0.561·11-s − 7.12·13-s − 0.561·15-s − 0.561·17-s − 19-s − 2.56·21-s − 3.12·23-s − 4.68·25-s − 27-s − 4·29-s + 5.12·31-s + 0.561·33-s + 1.43·35-s − 0.876·37-s + 7.12·39-s − 4·41-s − 3.68·43-s + 0.561·45-s + 8.56·47-s − 0.438·49-s + 0.561·51-s + 1.12·53-s − 0.315·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.251·5-s + 0.968·7-s + 0.333·9-s − 0.169·11-s − 1.97·13-s − 0.144·15-s − 0.136·17-s − 0.229·19-s − 0.558·21-s − 0.651·23-s − 0.936·25-s − 0.192·27-s − 0.742·29-s + 0.920·31-s + 0.0977·33-s + 0.243·35-s − 0.144·37-s + 1.14·39-s − 0.624·41-s − 0.561·43-s + 0.0837·45-s + 1.24·47-s − 0.0626·49-s + 0.0786·51-s + 0.154·53-s − 0.0425·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 0.561T + 11T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 0.876T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 - 8.56T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 7.43T + 61T^{2} \) |
| 67 | \( 1 + 6.87T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.561T + 73T^{2} \) |
| 79 | \( 1 + 5.12T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 1.12T + 89T^{2} \) |
| 97 | \( 1 - 5.36T + 97T^{2} \) |
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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
−8.909514643662694225947689517747, −7.86008619911119065979051289500, −7.42666071633604284635870947006, −6.43105412633197565083710562080, −5.48254544003759691038102975243, −4.88066885185544989411013225296, −4.10505277021321893083764564977, −2.57926815911539116939283741510, −1.70452623218217396883656710914, 0,
1.70452623218217396883656710914, 2.57926815911539116939283741510, 4.10505277021321893083764564977, 4.88066885185544989411013225296, 5.48254544003759691038102975243, 6.43105412633197565083710562080, 7.42666071633604284635870947006, 7.86008619911119065979051289500, 8.909514643662694225947689517747
