L(s) = 1 | − 1.41i·2-s − 3.19i·3-s − 0.0122·4-s − 4.52·6-s + 4.71i·7-s − 2.81i·8-s − 7.17·9-s − 5.53·11-s + 0.0389i·12-s + 0.648i·13-s + 6.68·14-s − 4.02·16-s − 1.05i·17-s + 10.1i·18-s − 7.88·19-s + ⋯ |
L(s) = 1 | − 1.00i·2-s − 1.84i·3-s − 0.00611·4-s − 1.84·6-s + 1.78i·7-s − 0.996i·8-s − 2.39·9-s − 1.66·11-s + 0.0112i·12-s + 0.179i·13-s + 1.78·14-s − 1.00·16-s − 0.256i·17-s + 2.40i·18-s − 1.80·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3672626138\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3672626138\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 - iT \) |
good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 3 | \( 1 + 3.19iT - 3T^{2} \) |
| 7 | \( 1 - 4.71iT - 7T^{2} \) |
| 11 | \( 1 + 5.53T + 11T^{2} \) |
| 13 | \( 1 - 0.648iT - 13T^{2} \) |
| 17 | \( 1 + 1.05iT - 17T^{2} \) |
| 19 | \( 1 + 7.88T + 19T^{2} \) |
| 23 | \( 1 + 3.05iT - 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 - 4.18T + 31T^{2} \) |
| 37 | \( 1 + 2.32iT - 37T^{2} \) |
| 41 | \( 1 + 4.06T + 41T^{2} \) |
| 43 | \( 1 - 4.14iT - 43T^{2} \) |
| 53 | \( 1 - 3.40iT - 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 + 8.40T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 3.21T + 71T^{2} \) |
| 73 | \( 1 - 2.87iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 15.9iT - 83T^{2} \) |
| 89 | \( 1 + 6.55T + 89T^{2} \) |
| 97 | \( 1 + 2.09iT - 97T^{2} \) |
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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
−8.840347089877969826670829176336, −8.369936882240558533885495763771, −7.51509361422142801413647823015, −6.43944639055879490811961759316, −6.03448351319758939383881902636, −4.88075028002780078950177929440, −2.85937727049917098445894072705, −2.50327176517297627110500130567, −1.79762450002978647854424281618, −0.13899908764331318779398840115,
2.62563021412960292194654653797, 3.77682299136418261185590046976, 4.60265981548442219256283531737, 5.21245486420860462436141568775, 6.19814375665637423786116564743, 7.16066181110970327867161812720, 8.136796223884603292610139401174, 8.521873842972241352533352525801, 9.902093145641637243750329279288, 10.46261359900107790941453176429