L(s) = 1 | + (0.133 − 1.99i)2-s − 4.89·3-s + (−3.96 − 0.533i)4-s + 1.12i·5-s + (−0.654 + 9.76i)6-s + 1.75i·7-s + (−1.59 + 7.83i)8-s + 14.9·9-s + (2.24 + 0.150i)10-s − 2.20·11-s + (19.3 + 2.61i)12-s − 6.00i·13-s + (3.49 + 0.234i)14-s − 5.49i·15-s + (15.4 + 4.23i)16-s + 22.4·17-s + ⋯ |
L(s) = 1 | + (0.0668 − 0.997i)2-s − 1.63·3-s + (−0.991 − 0.133i)4-s + 0.224i·5-s + (−0.109 + 1.62i)6-s + 0.250i·7-s + (−0.199 + 0.979i)8-s + 1.65·9-s + (0.224 + 0.0150i)10-s − 0.200·11-s + (1.61 + 0.217i)12-s − 0.461i·13-s + (0.249 + 0.0167i)14-s − 0.366i·15-s + (0.964 + 0.264i)16-s + 1.31·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.691264 - 0.0696195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.691264 - 0.0696195i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.133 + 1.99i)T \) |
| 23 | \( 1 - 4.79iT \) |
good | 3 | \( 1 + 4.89T + 9T^{2} \) |
| 5 | \( 1 - 1.12iT - 25T^{2} \) |
| 7 | \( 1 - 1.75iT - 49T^{2} \) |
| 11 | \( 1 + 2.20T + 121T^{2} \) |
| 13 | \( 1 + 6.00iT - 169T^{2} \) |
| 17 | \( 1 - 22.4T + 289T^{2} \) |
| 19 | \( 1 - 0.410T + 361T^{2} \) |
| 29 | \( 1 - 37.8iT - 841T^{2} \) |
| 31 | \( 1 - 47.4iT - 961T^{2} \) |
| 37 | \( 1 - 31.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.34T + 1.68e3T^{2} \) |
| 43 | \( 1 + 38.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.69iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 22.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 35.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 104.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 68.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 8.60T + 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 49.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 56.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 162.T + 9.40e3T^{2} \) |
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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
−12.28057791845275543833310052160, −11.37552928381623264116205783032, −10.55580917163302478221284601401, −9.962256813636697109920869802191, −8.488669527081469616640567100616, −6.93520972705126554537725313043, −5.56088940502477681307015341255, −4.96148490923388852304844616285, −3.26271303883650162390766091900, −1.13100368364016140150573265960,
0.63866119664744259091423337117, 4.11615526720849931252685347410, 5.20310376528560233028198639912, 5.99558159821619256900443503425, 6.97781613939514126065999910729, 8.020065031358563372798426260422, 9.514980619712828067485385294939, 10.39610996808336076052003850625, 11.58101522919193504067291012300, 12.42643823999084461866013774397