L(s) = 1 | + 4.53i·2-s − 12.5·4-s − 0.137·5-s + (14.6 + 11.3i)7-s − 20.6i·8-s − 0.623i·10-s + 10.1i·11-s − 15.1i·13-s + (−51.5 + 66.2i)14-s − 6.90·16-s + 88.2·17-s + 66.1i·19-s + 1.72·20-s − 45.9·22-s + 57.4i·23-s + ⋯ |
L(s) = 1 | + 1.60i·2-s − 1.56·4-s − 0.0122·5-s + (0.789 + 0.613i)7-s − 0.911i·8-s − 0.0197i·10-s + 0.277i·11-s − 0.323i·13-s + (−0.983 + 1.26i)14-s − 0.107·16-s + 1.25·17-s + 0.798i·19-s + 0.0192·20-s − 0.445·22-s + 0.520i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.439985213\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.439985213\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-14.6 - 11.3i)T \) |
good | 2 | \( 1 - 4.53iT - 8T^{2} \) |
| 5 | \( 1 + 0.137T + 125T^{2} \) |
| 11 | \( 1 - 10.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 15.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 88.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 66.1iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 57.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 118. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 307. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 441.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 93.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 311.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 321. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 574.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 732. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 63.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 621. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 95.2iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 850.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 140.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 775. iT - 9.12e5T^{2} \) |
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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
−10.78487258655487735678388036251, −9.715982880372877280783383104916, −8.797747218341702758838123236600, −7.990019458868039885853269507887, −7.48285276822974710100511697062, −6.34952580214392532244422780505, −5.45062483235747972606385055553, −4.92610273557942318396554958278, −3.46564230636380989331133193580, −1.64559548744094730481521441222,
0.44140947132868872315714750101, 1.53827479761919332466141902688, 2.66574775051687368205086119187, 3.86410753668787092417439363732, 4.57094135091581703645166549498, 5.83132272649843451555817191225, 7.29890678167479400688125152687, 8.190452252786179499199356248320, 9.262099521359822135813415179922, 10.01259794321280213640567127093