L(s) = 1 | + (0.869 − 1.11i)2-s + 3-s + (−0.489 − 1.93i)4-s + 1.54·5-s + (0.869 − 1.11i)6-s + 2.92·7-s + (−2.58 − 1.13i)8-s + 9-s + (1.34 − 1.72i)10-s + 4.46i·11-s + (−0.489 − 1.93i)12-s − 2.71i·13-s + (2.54 − 3.26i)14-s + 1.54·15-s + (−3.52 + 1.89i)16-s + 0.363i·17-s + ⋯ |
L(s) = 1 | + (0.614 − 0.788i)2-s + 0.577·3-s + (−0.244 − 0.969i)4-s + 0.689·5-s + (0.354 − 0.455i)6-s + 1.10·7-s + (−0.915 − 0.402i)8-s + 0.333·9-s + (0.423 − 0.544i)10-s + 1.34i·11-s + (−0.141 − 0.559i)12-s − 0.753i·13-s + (0.679 − 0.871i)14-s + 0.398·15-s + (−0.880 + 0.474i)16-s + 0.0881i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.22291 - 1.59411i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22291 - 1.59411i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.869 + 1.11i)T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + (3.23 + 3.53i)T \) |
good | 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 - 4.46iT - 11T^{2} \) |
| 13 | \( 1 + 2.71iT - 13T^{2} \) |
| 17 | \( 1 - 0.363iT - 17T^{2} \) |
| 19 | \( 1 - 1.20iT - 19T^{2} \) |
| 29 | \( 1 - 3.26iT - 29T^{2} \) |
| 31 | \( 1 + 1.19iT - 31T^{2} \) |
| 37 | \( 1 + 9.28T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 8.22iT - 43T^{2} \) |
| 47 | \( 1 - 2.10iT - 47T^{2} \) |
| 53 | \( 1 + 8.60T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 5.63T + 61T^{2} \) |
| 67 | \( 1 - 10.2iT - 67T^{2} \) |
| 71 | \( 1 + 6.80iT - 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 2.37T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 - 2.99iT - 89T^{2} \) |
| 97 | \( 1 + 13.6iT - 97T^{2} \) |
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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
−10.40191656832000943604750395691, −10.07099662572379673556260738772, −9.030096853997600449799167825892, −8.074707856574592298271125517539, −6.95032469533617347410660107374, −5.65055172707328310794888402833, −4.84116295908183989914353482491, −3.85596003978354399255823582785, −2.38518622034993666324356585077, −1.64495080816462194576567352857,
1.94498337781871337253826785995, 3.32852037599377670246220696523, 4.43606481958159079179840191093, 5.47316726877326157581967466105, 6.27543302699876661656015250166, 7.41397050979353331000988032033, 8.264166083614254019644669020181, 8.870027337087790399897525243489, 9.855277113888804304780012167125, 11.24062162436232628768070538675