L(s) = 1 | + (−1.82 + 2.38i)3-s + 7.37i·5-s + 2.64·7-s + (−2.35 − 8.68i)9-s + 2.61i·11-s + 6.35·13-s + (−17.5 − 13.4i)15-s + 12.1i·17-s − 10.2·19-s + (−4.82 + 6.30i)21-s + 4.30i·23-s − 29.4·25-s + (24.9 + 10.2i)27-s + 17.3i·29-s − 39.2·31-s + ⋯ |
L(s) = 1 | + (−0.607 + 0.794i)3-s + 1.47i·5-s + 0.377·7-s + (−0.261 − 0.965i)9-s + 0.237i·11-s + 0.488·13-s + (−1.17 − 0.896i)15-s + 0.714i·17-s − 0.538·19-s + (−0.229 + 0.300i)21-s + 0.187i·23-s − 1.17·25-s + (0.925 + 0.378i)27-s + 0.599i·29-s − 1.26·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6200060051\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6200060051\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.82 - 2.38i)T \) |
| 7 | \( 1 - 2.64T \) |
good | 5 | \( 1 - 7.37iT - 25T^{2} \) |
| 11 | \( 1 - 2.61iT - 121T^{2} \) |
| 13 | \( 1 - 6.35T + 169T^{2} \) |
| 17 | \( 1 - 12.1iT - 289T^{2} \) |
| 19 | \( 1 + 10.2T + 361T^{2} \) |
| 23 | \( 1 - 4.30iT - 529T^{2} \) |
| 29 | \( 1 - 17.3iT - 841T^{2} \) |
| 31 | \( 1 + 39.2T + 961T^{2} \) |
| 37 | \( 1 + 41.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 30.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 105. iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 41.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 20.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 27.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 67.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 60.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 63.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 89.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 63.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 19.1T + 9.40e3T^{2} \) |
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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
−10.27775222377672661700529049976, −9.271064531582704057614888784177, −8.413668462007374020642920375967, −7.30827256170813395807679953945, −6.57565318768758293949524196323, −5.88633731631096404032400102597, −4.91077069530276707392431282043, −3.82337895275984135860501124033, −3.18471736204843765778102241595, −1.78494252183075279278280317611,
0.20210862863536763185766739520, 1.20878404049214075260013077921, 2.16246405534583507265889001118, 3.83560987195402167273536169880, 4.98177090484056357443829449732, 5.36441666371487245042925911743, 6.37847323555336310134218421309, 7.28349987940619371465715291858, 8.248353132473927569727932741982, 8.639607515997426357271633394287