L(s) = 1 | + 5-s + 7-s − 6·11-s + 5·13-s + 6·17-s + 7·19-s − 6·23-s + 25-s + 3·29-s − 5·31-s + 35-s + 2·37-s + 3·41-s − 43-s + 12·47-s − 6·49-s − 6·53-s − 6·55-s − 12·59-s − 61-s + 5·65-s + 13·67-s + 12·71-s + 11·73-s − 6·77-s + 79-s + 6·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s + 1.38·13-s + 1.45·17-s + 1.60·19-s − 1.25·23-s + 1/5·25-s + 0.557·29-s − 0.898·31-s + 0.169·35-s + 0.328·37-s + 0.468·41-s − 0.152·43-s + 1.75·47-s − 6/7·49-s − 0.824·53-s − 0.809·55-s − 1.56·59-s − 0.128·61-s + 0.620·65-s + 1.58·67-s + 1.42·71-s + 1.28·73-s − 0.683·77-s + 0.112·79-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.864381634\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.864381634\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.24914053313696, −14.30144661245081, −14.03939645868522, −13.63441893442685, −13.04402794322057, −12.39735612290833, −12.07647358528022, −11.17214684171777, −10.82951234233050, −10.35824000199400, −9.567572885808898, −9.451884292477531, −8.305047077036514, −7.992479700796412, −7.679719594133990, −6.841654467484067, −6.015534449898223, −5.461765694949355, −5.325049206717101, −4.372040217791125, −3.510767181452206, −3.056133947743008, −2.240827003416992, −1.429285615065253, −0.6809468321044101,
0.6809468321044101, 1.429285615065253, 2.240827003416992, 3.056133947743008, 3.510767181452206, 4.372040217791125, 5.325049206717101, 5.461765694949355, 6.015534449898223, 6.841654467484067, 7.679719594133990, 7.992479700796412, 8.305047077036514, 9.451884292477531, 9.567572885808898, 10.35824000199400, 10.82951234233050, 11.17214684171777, 12.07647358528022, 12.39735612290833, 13.04402794322057, 13.63441893442685, 14.03939645868522, 14.30144661245081, 15.24914053313696