L(s) = 1 | + (−2 − 3.46i)5-s + (−3 − 1.73i)7-s + (−10.5 − 6.06i)11-s + (−11 − 19.0i)13-s + 11·17-s − 15.5i·19-s + (−21 + 12.1i)23-s + (4.50 − 7.79i)25-s + (−17 + 29.4i)29-s + (6 − 3.46i)31-s + 13.8i·35-s + 16·37-s + (6.5 + 11.2i)41-s + (43.5 + 25.1i)43-s + (−3 − 1.73i)47-s + ⋯ |
L(s) = 1 | + (−0.400 − 0.692i)5-s + (−0.428 − 0.247i)7-s + (−0.954 − 0.551i)11-s + (−0.846 − 1.46i)13-s + 0.647·17-s − 0.820i·19-s + (−0.913 + 0.527i)23-s + (0.180 − 0.311i)25-s + (−0.586 + 1.01i)29-s + (0.193 − 0.111i)31-s + 0.395i·35-s + 0.432·37-s + (0.158 + 0.274i)41-s + (1.01 + 0.584i)43-s + (−0.0638 − 0.0368i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03360789406\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03360789406\) |
\(L(2)\) |
|
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 | \( 1 + (2 + 3.46i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (3 + 1.73i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (10.5 + 6.06i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (11 + 19.0i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 11T + 289T^{2} \) |
| 19 | \( 1 + 15.5iT - 361T^{2} \) |
| 23 | \( 1 + (21 - 12.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (17 - 29.4i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-6 + 3.46i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-6.5 - 11.2i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-43.5 - 25.1i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (3 + 1.73i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 52T + 2.80e3T^{2} \) |
| 59 | \( 1 + (46.5 - 26.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (8 - 13.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (100.5 - 58.0i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-24 - 13.8i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-30 - 17.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-21.5 + 37.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−8.468191568087287407188939165083, −7.80076788899631851201528854394, −7.26573937545408973156480706794, −5.94561633442852632117025894485, −5.33178223739015454862309554262, −4.51067107564727874572728275457, −3.35198568624860014031444233188, −2.60932577569916461031417806774, −0.903295718053942047716103165871, −0.01073357050173477981070834141,
1.93993941093097408317467184826, 2.75204314666617265797526472566, 3.84881689775753165349156902381, 4.65944755376558780574953192055, 5.75868376879623558262592751266, 6.52200343859060624362387762216, 7.48445677265103343317781308256, 7.78826749451571331312181862325, 9.025968993596972762800681126405, 9.744045797567740565438683186796