L(s) = 1 | + 2i·2-s + 4.69i·3-s − 4·4-s + (−1.54 − 11.0i)5-s − 9.38·6-s − 8i·8-s + 4.97·9-s + (22.1 − 3.08i)10-s + 1.37·11-s − 18.7i·12-s + 69.4i·13-s + (51.9 − 7.23i)15-s + 16·16-s − 26.3i·17-s + 9.94i·18-s + 14.1·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.903i·3-s − 0.5·4-s + (−0.137 − 0.990i)5-s − 0.638·6-s − 0.353i·8-s + 0.184·9-s + (0.700 − 0.0975i)10-s + 0.0377·11-s − 0.451i·12-s + 1.48i·13-s + (0.894 − 0.124i)15-s + 0.250·16-s − 0.375i·17-s + 0.130i·18-s + 0.171·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.050423015\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050423015\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (1.54 + 11.0i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 4.69iT - 27T^{2} \) |
| 11 | \( 1 - 1.37T + 1.33e3T^{2} \) |
| 13 | \( 1 - 69.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 26.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 14.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 75.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 245.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 245. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 238.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 194. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 375. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 511. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 656.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 809.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 220. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 898. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 708.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 419. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 152. iT - 9.12e5T^{2} \) |
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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
−10.94992896750539729038576496814, −9.630967609316056601376216288797, −9.374408776515234599531628723291, −8.448232162456548391066506103373, −7.39961990192963496910722037434, −6.37924141693607419325317406486, −5.08980797712833094666782116910, −4.56396476246376082964560760954, −3.61803550272443179172674083990, −1.49433800779627892859452900938,
0.33604575836454291714440573453, 1.76729813172289085050901146135, 2.87721404428721030572583360867, 3.87419177298880095579422969958, 5.41841655454780255644117341392, 6.46722421418250514262696077182, 7.45646750989710516233148148455, 8.067618802261345587059713046449, 9.332880318685601522726602750403, 10.53270884839147377006766905138