| L(s) = 1 | + 0.746·2-s − 1.44·4-s + 2.36·5-s − 0.386·7-s − 2.56·8-s + 1.76·10-s + 5.22·11-s − 1.64·13-s − 0.288·14-s + 0.968·16-s − 6.42·17-s − 3.07·19-s − 3.41·20-s + 3.90·22-s − 7.77·23-s + 0.584·25-s − 1.22·26-s + 0.557·28-s − 9.40·29-s − 2.39·31-s + 5.86·32-s − 4.79·34-s − 0.912·35-s + 4.99·37-s − 2.29·38-s − 6.07·40-s − 41-s + ⋯ |
| L(s) = 1 | + 0.527·2-s − 0.721·4-s + 1.05·5-s − 0.145·7-s − 0.908·8-s + 0.557·10-s + 1.57·11-s − 0.456·13-s − 0.0770·14-s + 0.242·16-s − 1.55·17-s − 0.706·19-s − 0.762·20-s + 0.831·22-s − 1.62·23-s + 0.116·25-s − 0.240·26-s + 0.105·28-s − 1.74·29-s − 0.429·31-s + 1.03·32-s − 0.822·34-s − 0.154·35-s + 0.820·37-s − 0.372·38-s − 0.960·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3321 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.746T + 2T^{2} \) |
| 5 | \( 1 - 2.36T + 5T^{2} \) |
| 7 | \( 1 + 0.386T + 7T^{2} \) |
| 11 | \( 1 - 5.22T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 + 3.07T + 19T^{2} \) |
| 23 | \( 1 + 7.77T + 23T^{2} \) |
| 29 | \( 1 + 9.40T + 29T^{2} \) |
| 31 | \( 1 + 2.39T + 31T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 43 | \( 1 - 5.60T + 43T^{2} \) |
| 47 | \( 1 - 5.66T + 47T^{2} \) |
| 53 | \( 1 - 1.07T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 - 9.57T + 61T^{2} \) |
| 67 | \( 1 + 5.57T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 0.777T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
| 97 | \( 1 + 7.83T + 97T^{2} \) |
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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
−8.508901043708864008430245565753, −7.40755825083767462071083074020, −6.32166600042375494756892973086, −6.12346353747562534546145752111, −5.22670180886858124393916565338, −4.12609145465500233063964121282, −3.95842997163901490013710290623, −2.48322898290122188191055193068, −1.67878947232721866792263909552, 0,
1.67878947232721866792263909552, 2.48322898290122188191055193068, 3.95842997163901490013710290623, 4.12609145465500233063964121282, 5.22670180886858124393916565338, 6.12346353747562534546145752111, 6.32166600042375494756892973086, 7.40755825083767462071083074020, 8.508901043708864008430245565753
