L(s) = 1 | − 1.41·2-s + 2.00·4-s + 4.89i·5-s − 2.82·8-s − 6.92i·10-s + 16.9·11-s + 1.73i·13-s + 4.00·16-s + 4.89i·17-s − 29.4i·19-s + 9.79i·20-s − 24·22-s + 8.48·23-s + 1.00·25-s − 2.44i·26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.500·4-s + 0.979i·5-s − 0.353·8-s − 0.692i·10-s + 1.54·11-s + 0.133i·13-s + 0.250·16-s + 0.288i·17-s − 1.54i·19-s + 0.489i·20-s − 1.09·22-s + 0.368·23-s + 0.0400·25-s − 0.0942i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.494681108\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.494681108\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 4.89iT - 25T^{2} \) |
| 11 | \( 1 - 16.9T + 121T^{2} \) |
| 13 | \( 1 - 1.73iT - 169T^{2} \) |
| 17 | \( 1 - 4.89iT - 289T^{2} \) |
| 19 | \( 1 + 29.4iT - 361T^{2} \) |
| 23 | \( 1 - 8.48T + 529T^{2} \) |
| 29 | \( 1 - 33.9T + 841T^{2} \) |
| 31 | \( 1 + 12.1iT - 961T^{2} \) |
| 37 | \( 1 + 47T + 1.36e3T^{2} \) |
| 41 | \( 1 - 68.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 31T + 1.84e3T^{2} \) |
| 47 | \( 1 + 83.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 76.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 83.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 31T + 4.48e3T^{2} \) |
| 71 | \( 1 + 59.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 81.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 41T + 6.24e3T^{2} \) |
| 83 | \( 1 + 4.89iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 58.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 41.5iT - 9.40e3T^{2} \) |
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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
−10.09798937946665718368681464222, −9.062434595354888166078292361753, −8.633646707164171569919290065306, −7.24428668542179201986882281219, −6.83549312817224574484701577047, −6.06642268076916765132650969572, −4.60781146643723200598155846865, −3.42516704491173606144146485569, −2.42124336308246298821648205328, −1.00679076852491754895756017456,
0.837678439018601728425428380588, 1.75466802421105957384220217213, 3.40264655621862162388385200684, 4.44050611416465110061971702525, 5.56502647622743463685627132144, 6.49878876514486905018689092524, 7.39004119215251055501679772811, 8.479386800132230718810269209059, 8.884891324571051146717925361367, 9.699758950908859196373894170474