L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 12-s − 2·13-s + 14-s + 15-s + 16-s + 6·17-s + 18-s − 4·19-s − 20-s − 21-s − 24-s + 25-s − 2·26-s − 27-s + 28-s + 6·29-s + 30-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.362137048\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.362137048\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.88283988497045, −12.51313332372866, −12.17076954512450, −11.76438255479530, −11.25951534169614, −10.87579957051638, −10.28793775448614, −9.963512188038810, −9.421520217979040, −8.658616246950856, −8.053215461639827, −7.892421213999907, −7.146481182271716, −6.808128163703953, −6.128792097180144, −5.838832041886525, −5.097961479148401, −4.690191484505234, −4.453345772931352, −3.559952989546152, −3.236612501806748, −2.560048797671794, −1.770666605077580, −1.266378821782842, −0.4040185664566930,
0.4040185664566930, 1.266378821782842, 1.770666605077580, 2.560048797671794, 3.236612501806748, 3.559952989546152, 4.453345772931352, 4.690191484505234, 5.097961479148401, 5.838832041886525, 6.128792097180144, 6.808128163703953, 7.146481182271716, 7.892421213999907, 8.053215461639827, 8.658616246950856, 9.421520217979040, 9.963512188038810, 10.28793775448614, 10.87579957051638, 11.25951534169614, 11.76438255479530, 12.17076954512450, 12.51313332372866, 12.88283988497045