| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 12-s − 2·13-s − 14-s + 16-s + 6·17-s + 18-s + 8·19-s + 21-s − 24-s − 2·26-s − 27-s − 28-s + 6·29-s − 4·31-s + 32-s + 6·34-s + 36-s + 10·37-s + 8·38-s + 2·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s + 0.218·21-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.64·37-s + 1.29·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.104878897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104878897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−19.63033268246537, −18.47210488676457, −18.22300602320791, −16.98370370828547, −16.64278153346950, −15.88024546438669, −15.24145887179108, −14.27963989845924, −13.83660992895534, −12.85389653218306, −12.18783899491632, −11.74921594511324, −10.81417009341772, −9.944565703326023, −9.406502048415156, −7.910500082098510, −7.331678879228486, −6.349919672635291, −5.519149339987972, −4.879978618529908, −3.669650669553365, −2.775761760115944, −1.109519981333222,
1.109519981333222, 2.775761760115944, 3.669650669553365, 4.879978618529908, 5.519149339987972, 6.349919672635291, 7.331678879228486, 7.910500082098510, 9.406502048415156, 9.944565703326023, 10.81417009341772, 11.74921594511324, 12.18783899491632, 12.85389653218306, 13.83660992895534, 14.27963989845924, 15.24145887179108, 15.88024546438669, 16.64278153346950, 16.98370370828547, 18.22300602320791, 18.47210488676457, 19.63033268246537
