L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 3·17-s − 18-s − 19-s − 20-s + 21-s + 22-s − 4·23-s + 24-s + 25-s − 26-s − 27-s − 28-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 − 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.195097656\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.195097656\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.42201272695698, −14.13197270242777, −13.30734668264052, −12.83454154544508, −12.25887464545498, −11.91901055234220, −11.34718514535702, −10.85541780725482, −10.24887887309185, −9.985897892873667, −9.332396346251407, −8.728696273012764, −8.098473524422322, −7.754321915259360, −7.113201197621286, −6.557741814173188, −5.932373361946246, −5.615123387848661, −4.649127636794165, −4.186694326056721, −3.416806579321631, −2.744216468455401, −2.050024994186882, −1.009749090747642, −0.5502549141172865,
0.5502549141172865, 1.009749090747642, 2.050024994186882, 2.744216468455401, 3.416806579321631, 4.186694326056721, 4.649127636794165, 5.615123387848661, 5.932373361946246, 6.557741814173188, 7.113201197621286, 7.754321915259360, 8.098473524422322, 8.728696273012764, 9.332396346251407, 9.985897892873667, 10.24887887309185, 10.85541780725482, 11.34718514535702, 11.91901055234220, 12.25887464545498, 12.83454154544508, 13.30734668264052, 14.13197270242777, 14.42201272695698