L(s) = 1 | + (−2.5 + 4.33i)5-s + (8 + 13.8i)7-s + (30 + 51.9i)11-s + (−43 + 74.4i)13-s + 18·17-s + 44·19-s + (−24 + 41.5i)23-s + (−12.5 − 21.6i)25-s + (93 + 161. i)29-s + (−88 + 152. i)31-s − 80·35-s + 254·37-s + (−93 + 161. i)41-s + (50 + 86.6i)43-s + (−84 − 145. i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.431 + 0.748i)7-s + (0.822 + 1.42i)11-s + (−0.917 + 1.58i)13-s + 0.256·17-s + 0.531·19-s + (−0.217 + 0.376i)23-s + (−0.100 − 0.173i)25-s + (0.595 + 1.03i)29-s + (−0.509 + 0.883i)31-s − 0.386·35-s + 1.12·37-s + (−0.354 + 0.613i)41-s + (0.177 + 0.307i)43-s + (−0.260 − 0.451i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.853609839\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.853609839\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 7 | \( 1 + (-8 - 13.8i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-30 - 51.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (43 - 74.4i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 18T + 4.91e3T^{2} \) |
| 19 | \( 1 - 44T + 6.85e3T^{2} \) |
| 23 | \( 1 + (24 - 41.5i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-93 - 161. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (88 - 152. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 - 254T + 5.06e4T^{2} \) |
| 41 | \( 1 + (93 - 161. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-50 - 86.6i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (84 + 145. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 498T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-126 + 218. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-29 - 50.2i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-518 + 897. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 168T + 3.57e5T^{2} \) |
| 73 | \( 1 - 506T + 3.89e5T^{2} \) |
| 79 | \( 1 + (136 + 235. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (474 + 820. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-383 - 663. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−9.497586205429292928245076863332, −8.699933060971299745678436256583, −7.64555179667562509002798107015, −6.98955959493908023283450920416, −6.36714415505822392502056569610, −5.04330283727452975346653873661, −4.56271217361229239488727012135, −3.44839664020609712214412622713, −2.22207379502385187622158544115, −1.53297698433919857716908294045,
0.46210142232394493575342923117, 1.02762245560331568968348159356, 2.63686675039860472669532353469, 3.60725250117881710890548076677, 4.43281296157921814799798921722, 5.45064803218611898026405339361, 6.09592068770082499213740821250, 7.27620411286742833609475358593, 7.957109355746499009937767322974, 8.467532787410794763303008560621