| L(s) = 1 | − 2-s + 0.445·3-s + 4-s + 1.24·5-s − 0.445·6-s + 1.80·7-s − 8-s − 0.801·9-s − 1.24·10-s − 1.24·11-s + 0.445·12-s − 1.80·13-s − 1.80·14-s + 0.554·15-s + 16-s − 0.445·17-s + 0.801·18-s + 1.24·20-s + 0.801·21-s + 1.24·22-s − 0.445·24-s + 0.554·25-s + 1.80·26-s − 0.801·27-s + 1.80·28-s − 0.554·30-s − 32-s + ⋯ |
| L(s) = 1 | − 2-s + 0.445·3-s + 4-s + 1.24·5-s − 0.445·6-s + 1.80·7-s − 8-s − 0.801·9-s − 1.24·10-s − 1.24·11-s + 0.445·12-s − 1.80·13-s − 1.80·14-s + 0.554·15-s + 16-s − 0.445·17-s + 0.801·18-s + 1.24·20-s + 0.801·21-s + 1.24·22-s − 0.445·24-s + 0.554·25-s + 1.80·26-s − 0.801·27-s + 1.80·28-s − 0.554·30-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7560953902\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7560953902\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 101 | \( 1 - T \) |
| good | 3 | \( 1 - 0.445T + T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - 1.80T + T^{2} \) |
| 11 | \( 1 + 1.24T + T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 + 0.445T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.80T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 0.445T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.24T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−11.22850582360781448332109446705, −10.40387214487429049529103288266, −9.661074496296445972701218916501, −8.673490439319629927427318148148, −7.989891165068875572869359213590, −7.18457248385764020162496399326, −5.65161334981991669304165218273, −5.00937812366992867608066973067, −2.55420776607694329291774055831, −2.03811030715590218381504477610,
2.03811030715590218381504477610, 2.55420776607694329291774055831, 5.00937812366992867608066973067, 5.65161334981991669304165218273, 7.18457248385764020162496399326, 7.989891165068875572869359213590, 8.673490439319629927427318148148, 9.661074496296445972701218916501, 10.40387214487429049529103288266, 11.22850582360781448332109446705
