L(s) = 1 | + (2 − 2i)2-s + (−2.23 + 2.23i)3-s + 8.94i·6-s + (2.67 − 18.3i)7-s + (16 + 16i)8-s + 17i·9-s + 24·11-s + (−42.4 + 42.4i)13-s + (−31.3 − 42i)14-s + 64·16-s + (91.6 + 91.6i)17-s + (34 + 34i)18-s + 80.4·19-s + (35 + 46.9i)21-s + (48 − 48i)22-s + (59 + 59i)23-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.430 + 0.430i)3-s + 0.608i·6-s + (0.144 − 0.989i)7-s + (0.707 + 0.707i)8-s + 0.629i·9-s + 0.657·11-s + (−0.906 + 0.906i)13-s + (−0.597 − 0.801i)14-s + 16-s + (1.30 + 1.30i)17-s + (0.445 + 0.445i)18-s + 0.971·19-s + (0.363 + 0.487i)21-s + (0.465 − 0.465i)22-s + (0.534 + 0.534i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.20384 + 0.456729i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.20384 + 0.456729i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-2.67 + 18.3i)T \) |
good | 2 | \( 1 + (-2 + 2i)T - 8iT^{2} \) |
| 3 | \( 1 + (2.23 - 2.23i)T - 27iT^{2} \) |
| 11 | \( 1 - 24T + 1.33e3T^{2} \) |
| 13 | \( 1 + (42.4 - 42.4i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-91.6 - 91.6i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 - 80.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-59 - 59i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 52iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 58.1iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (93 - 93i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 26.8iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (221 + 221i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (51.4 + 51.4i)T + 1.03e5iT^{2} \) |
| 53 | \( 1 + (-87 - 87i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 420.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 339. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + (-277 + 277i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 916T + 3.57e5T^{2} \) |
| 73 | \( 1 + (78.2 - 78.2i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 478iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-462. + 462. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 4.47T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-959. - 959. i)T + 9.12e5iT^{2} \) |
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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
−12.10560950759206661893627967534, −11.50265117644153074573268956800, −10.53592891659627835708974156082, −9.759687351425455187791383101584, −8.074515501334951076682189364634, −7.14184562161796785315440842483, −5.41998588573014733492031613937, −4.42224252059760775132197072062, −3.50004788490674053173352356656, −1.66361304469990004579170472372,
0.980874808875993978604373887776, 3.17017921739595577511774286506, 5.04164015106336359236003429223, 5.63429173100620676129578373683, 6.75968893520848310087545576382, 7.61743915075636610899737199546, 9.204233418610638340522896126512, 10.07753720304788639092387939569, 11.67431265391238860856675466558, 12.23242259383910226408303228106