L(s) = 1 | − 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (−0.120 + 4.99i)5-s − 2.44·6-s − 0.825·7-s + 2.82i·8-s − 2.99·9-s + (7.06 + 0.170i)10-s − 20.1i·11-s + 3.46i·12-s + 8.65i·13-s + 1.16i·14-s + (8.65 + 0.208i)15-s + 4.00·16-s + 10.6·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (−0.0241 + 0.999i)5-s − 0.408·6-s − 0.117·7-s + 0.353i·8-s − 0.333·9-s + (0.706 + 0.0170i)10-s − 1.82i·11-s + 0.288i·12-s + 0.665i·13-s + 0.0834i·14-s + (0.577 + 0.0139i)15-s + 0.250·16-s + 0.626·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.867 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4830924846\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4830924846\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + (0.120 - 4.99i)T \) |
| 23 | \( 1 + (10.9 - 20.2i)T \) |
good | 7 | \( 1 + 0.825T + 49T^{2} \) |
| 11 | \( 1 + 20.1iT - 121T^{2} \) |
| 13 | \( 1 - 8.65iT - 169T^{2} \) |
| 17 | \( 1 - 10.6T + 289T^{2} \) |
| 19 | \( 1 + 23.3iT - 361T^{2} \) |
| 29 | \( 1 + 27.2T + 841T^{2} \) |
| 31 | \( 1 + 3.63T + 961T^{2} \) |
| 37 | \( 1 + 21.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 56.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 37.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + 37.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 19.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 20.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 36.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 43.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 76.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 73.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 74.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 115.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 85.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 117.T + 9.40e3T^{2} \) |
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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
−9.889867033585891766646748151630, −8.947748537446351809751450537703, −8.099809931294265049316134026911, −7.08706534401082173510135278504, −6.22591506845268141711834809414, −5.29730971502545485247346641630, −3.62579979007288294745960173755, −3.05269651599744303510288653344, −1.75347539373637794562155278392, −0.16783066214608974369543701877,
1.72091802308372803039932191695, 3.59660945293452341683622160301, 4.58770025674402638583531449735, 5.23556538186575133939841215900, 6.19973717693273119390581961600, 7.45667820900459644357694110752, 8.114053450583101600129881614383, 9.023285093711395931947165585774, 9.950113066953324677406434877324, 10.22849865943199208574465173938