L(s) = 1 | − 0.112·2-s − 1.73i·3-s − 3.98·4-s − 2.23i·5-s + 0.195i·6-s + (−6.71 + 1.98i)7-s + 0.902·8-s − 2.99·9-s + 0.252i·10-s − 15.8·11-s + 6.90i·12-s − 13.3i·13-s + (0.758 − 0.224i)14-s − 3.87·15-s + 15.8·16-s + 15.6i·17-s + ⋯ |
L(s) = 1 | − 0.0564·2-s − 0.577i·3-s − 0.996·4-s − 0.447i·5-s + 0.0326i·6-s + (−0.959 + 0.283i)7-s + 0.112·8-s − 0.333·9-s + 0.0252i·10-s − 1.44·11-s + 0.575i·12-s − 1.02i·13-s + (0.0541 − 0.0160i)14-s − 0.258·15-s + 0.990·16-s + 0.922i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0583761 - 0.403538i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0583761 - 0.403538i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (6.71 - 1.98i)T \) |
good | 2 | \( 1 + 0.112T + 4T^{2} \) |
| 11 | \( 1 + 15.8T + 121T^{2} \) |
| 13 | \( 1 + 13.3iT - 169T^{2} \) |
| 17 | \( 1 - 15.6iT - 289T^{2} \) |
| 19 | \( 1 + 30.8iT - 361T^{2} \) |
| 23 | \( 1 - 3.63T + 529T^{2} \) |
| 29 | \( 1 - 14.5T + 841T^{2} \) |
| 31 | \( 1 + 11.3iT - 961T^{2} \) |
| 37 | \( 1 - 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 27.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 12.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 80.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 79.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 94.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 103.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 113.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 20.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 1.27T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 131. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 12.4iT - 9.40e3T^{2} \) |
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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
−12.90530227988174125794544303223, −12.63080798498657497454641160542, −10.80126556816310096615567075991, −9.697362898276254421781582194490, −8.619879636700383674713616256321, −7.69271872363467094276096981207, −6.01095975706788352016015363332, −4.88751123307868311969223732529, −2.98845106546030290889479681782, −0.30646226579973574785856289536,
3.16369221078375360571947733378, 4.50033305824677525352445900492, 5.84912239384846311514974660937, 7.44300582144029800501847600718, 8.764900119862718863073434959475, 9.910688397786543126585667491203, 10.38270370717940827091478413134, 11.95396304852745266873048384748, 13.18714947771442294146899514338, 13.90971138260856678632651288319