L(s) = 1 | + (−0.454 − 0.787i)5-s + (6.10 + 3.52i)7-s + (6.96 + 4.02i)11-s + (−3.35 − 5.81i)13-s + 26.3·17-s − 20.5i·19-s + (−21.8 + 12.6i)23-s + (12.0 − 20.9i)25-s + (15.1 − 26.2i)29-s + (0.120 − 0.0693i)31-s − 6.41i·35-s − 69.7·37-s + (29.3 + 50.8i)41-s + (2.45 + 1.41i)43-s + (70.7 + 40.8i)47-s + ⋯ |
L(s) = 1 | + (−0.0909 − 0.157i)5-s + (0.872 + 0.503i)7-s + (0.633 + 0.365i)11-s + (−0.258 − 0.447i)13-s + 1.54·17-s − 1.08i·19-s + (−0.949 + 0.547i)23-s + (0.483 − 0.837i)25-s + (0.523 − 0.906i)29-s + (0.00387 − 0.00223i)31-s − 0.183i·35-s − 1.88·37-s + (0.716 + 1.24i)41-s + (0.0571 + 0.0330i)43-s + (1.50 + 0.869i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.398434099\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.398434099\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.454 + 0.787i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (-6.10 - 3.52i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-6.96 - 4.02i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (3.35 + 5.81i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 26.3T + 289T^{2} \) |
| 19 | \( 1 + 20.5iT - 361T^{2} \) |
| 23 | \( 1 + (21.8 - 12.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-15.1 + 26.2i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-0.120 + 0.0693i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 69.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-29.3 - 50.8i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-2.45 - 1.41i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-70.7 - 40.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 30.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-77.1 + 44.5i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (24.0 - 41.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-44.0 + 25.4i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 68.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 22.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-34.4 - 19.8i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-23.0 - 13.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 25.7T + 7.92e3T^{2} \) |
| 97 | \( 1 + (52.3 - 90.7i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.100124882416186360926239027452, −8.195441378685967514416322858893, −7.72701306874173326434058534604, −6.72480398930717360977357443265, −5.75773709939066736909783543728, −5.01787997507352442479348595804, −4.20996335736799604204829608928, −3.05363534258135660595953647276, −1.97485415215748337525122675331, −0.819305864787609040396363430685,
0.989742789447545059699184917652, 1.92353586779068660647735750622, 3.40597063652790024485229911762, 4.03123290656549247977587588453, 5.13706381258813632505358111496, 5.84279682400070566089845992875, 6.95498507688203610532079267044, 7.54093675703952146232331542018, 8.379703095633025626463293315695, 9.040219618607722210195370714080