L(s) = 1 | − 3.74·2-s + 3.97i·3-s + 10.0·4-s − 6.95i·5-s − 14.8i·6-s + (5.95 + 3.68i)7-s − 22.5·8-s − 6.77·9-s + 26.0i·10-s + 12.0·11-s + 39.8i·12-s + 3.60i·13-s + (−22.2 − 13.8i)14-s + 27.6·15-s + 44.4·16-s + 20.4i·17-s + ⋯ |
L(s) = 1 | − 1.87·2-s + 1.32i·3-s + 2.50·4-s − 1.39i·5-s − 2.47i·6-s + (0.850 + 0.526i)7-s − 2.82·8-s − 0.753·9-s + 2.60i·10-s + 1.09·11-s + 3.31i·12-s + 0.277i·13-s + (−1.59 − 0.986i)14-s + 1.84·15-s + 2.77·16-s + 1.20i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.571143 + 0.317989i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.571143 + 0.317989i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-5.95 - 3.68i)T \) |
| 13 | \( 1 - 3.60iT \) |
good | 2 | \( 1 + 3.74T + 4T^{2} \) |
| 3 | \( 1 - 3.97iT - 9T^{2} \) |
| 5 | \( 1 + 6.95iT - 25T^{2} \) |
| 11 | \( 1 - 12.0T + 121T^{2} \) |
| 17 | \( 1 - 20.4iT - 289T^{2} \) |
| 19 | \( 1 - 13.2iT - 361T^{2} \) |
| 23 | \( 1 - 1.12T + 529T^{2} \) |
| 29 | \( 1 - 13.3T + 841T^{2} \) |
| 31 | \( 1 + 36.5iT - 961T^{2} \) |
| 37 | \( 1 - 5.05T + 1.36e3T^{2} \) |
| 41 | \( 1 + 0.105iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 69.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 32.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 69.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 87.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 73.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 92.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + 17.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 97.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 55.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 55.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 42.3iT - 9.40e3T^{2} \) |
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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
−14.64629453523802543745650981076, −12.35681656458315109196166261902, −11.45204974006473852415189017798, −10.41361184648070166243746935877, −9.310633243392816474756065740927, −8.873271691040229494822462874346, −7.941231208341233406846540519435, −5.92190922759162977972060075980, −4.28406596508310059501982846962, −1.52264251321747590348478188265,
1.12795501573305902259643148446, 2.62423665005319617452749663882, 6.51827784376458440282204269041, 7.11503842954492152210482110333, 7.80222977130727098011298520023, 9.114313065049879387859982589670, 10.46463389678389322589075171295, 11.27802830667744659786060794895, 12.03106476217844834101198862814, 13.87729998998747320018737905681