L(s) = 1 | − i·2-s − i·3-s − 4-s − 2.84i·5-s − 6-s + i·8-s − 9-s − 2.84·10-s + (2.48 + 2.19i)11-s + i·12-s + 0.726·13-s − 2.84·15-s + 16-s − 5.60·17-s + i·18-s − 6.09·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 1.27i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 0.899·10-s + (0.749 + 0.662i)11-s + 0.288i·12-s + 0.201·13-s − 0.734·15-s + 0.250·16-s − 1.35·17-s + 0.235i·18-s − 1.39·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3885278307\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3885278307\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-2.48 - 2.19i)T \) |
good | 5 | \( 1 + 2.84iT - 5T^{2} \) |
| 13 | \( 1 - 0.726T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 6.09T + 19T^{2} \) |
| 23 | \( 1 - 2.98T + 23T^{2} \) |
| 29 | \( 1 - 3.95iT - 29T^{2} \) |
| 31 | \( 1 - 10.0iT - 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 + 6.88T + 41T^{2} \) |
| 43 | \( 1 - 5.82iT - 43T^{2} \) |
| 47 | \( 1 + 6.81iT - 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 - 12.4iT - 59T^{2} \) |
| 61 | \( 1 + 1.34T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 1.05T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 8.51iT - 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 4.47iT - 89T^{2} \) |
| 97 | \( 1 + 4.27iT - 97T^{2} \) |
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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
−8.701681771350290538349013706633, −8.489747711549021778067260898473, −7.11922894453544292036900761348, −6.63842790603606103443357201807, −5.54398429098858805556647632064, −4.65616498485999462123958665018, −4.24798273114793556384624469867, −3.02064439435748523155022110827, −1.83660333431070211987231689287, −1.28656137549361924704856962025,
0.11940734200354774937074580390, 2.11648130361968479559109077687, 3.13413381545582066961436658628, 4.00039113681630247717844904734, 4.58982209136959979114419831249, 5.84938324862691619108068395829, 6.35694443088457015039330625847, 6.87727882057530750152074777170, 7.76884463518532479047465349536, 8.705849541276141064596207894468