L(s) = 1 | + (−1.30 − 0.535i)2-s + (1.42 + 1.40i)4-s + 2.23·5-s + (−1.11 − 2.59i)8-s + (−2.92 − 1.19i)10-s + (0.0729 + 3.99i)16-s − 7.33i·17-s + 0.408i·19-s + (3.19 + 3.13i)20-s − 9.47i·23-s + 5.00·25-s − 2.70·31-s + (2.04 − 5.27i)32-s + (−3.92 + 9.60i)34-s + (0.218 − 0.535i)38-s + ⋯ |
L(s) = 1 | + (−0.925 − 0.378i)2-s + (0.713 + 0.700i)4-s + 0.999·5-s + (−0.395 − 0.918i)8-s + (−0.925 − 0.378i)10-s + (0.0182 + 0.999i)16-s − 1.77i·17-s + 0.0938i·19-s + (0.713 + 0.700i)20-s − 1.97i·23-s + 1.00·25-s − 0.486·31-s + (0.361 − 0.932i)32-s + (−0.673 + 1.64i)34-s + (0.0355 − 0.0868i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.209048004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.209048004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.30 + 0.535i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.33iT - 17T^{2} \) |
| 19 | \( 1 - 0.408iT - 19T^{2} \) |
| 23 | \( 1 + 9.47iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2.70T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6.01iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.638299470237975054561117815117, −9.099673461460769792897307876634, −8.308509020714114846647059087576, −7.21717064644995586801745815483, −6.60452342635879106567410460824, −5.57665955001465431013569630619, −4.42841263947727858223706910973, −2.93104795586752939732754221695, −2.22987556260475821178623666782, −0.78227655326865044187879551347,
1.36011799451730205227434958596, 2.26262389082652124373785055780, 3.72338355480310270303621981423, 5.35744749503837646152795583964, 5.82590154948876326366507750189, 6.75212070341228236817742645547, 7.56882264889262893763251841999, 8.557226696783962242838310731480, 9.139841678931265609450096751345, 10.03022090007833616765943288672