L(s) = 1 | + (−1.11 − 1.32i)3-s + (0.337 − 0.195i)5-s + (−1.39 + 2.24i)7-s + (−0.529 + 2.95i)9-s + (0.748 − 1.29i)11-s + 3.28·13-s + (−0.634 − 0.232i)15-s + (−1.68 + 2.91i)17-s + (2.56 + 4.43i)19-s + (4.53 − 0.644i)21-s + (4.72 − 2.72i)23-s + (−2.42 + 4.19i)25-s + (4.51 − 2.57i)27-s − 4.13·29-s + (3.60 + 2.07i)31-s + ⋯ |
L(s) = 1 | + (−0.641 − 0.766i)3-s + (0.151 − 0.0872i)5-s + (−0.527 + 0.849i)7-s + (−0.176 + 0.984i)9-s + (0.225 − 0.390i)11-s + 0.911·13-s + (−0.163 − 0.0599i)15-s + (−0.407 + 0.706i)17-s + (0.587 + 1.01i)19-s + (0.990 − 0.140i)21-s + (0.985 − 0.569i)23-s + (−0.484 + 0.839i)25-s + (0.868 − 0.496i)27-s − 0.768·29-s + (0.647 + 0.373i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14343 + 0.149280i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14343 + 0.149280i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.11 + 1.32i)T \) |
| 7 | \( 1 + (1.39 - 2.24i)T \) |
good | 5 | \( 1 + (-0.337 + 0.195i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.748 + 1.29i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + (1.68 - 2.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.56 - 4.43i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.72 + 2.72i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + (-3.60 - 2.07i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.46 + 4.31i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 4.79iT - 43T^{2} \) |
| 47 | \( 1 + (-2.51 - 4.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.499 - 0.864i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.36 + 0.785i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.40 - 5.90i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.05 + 1.76i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.3iT - 71T^{2} \) |
| 73 | \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.239 - 0.414i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 + (-2.54 - 4.41i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 9.00iT - 97T^{2} \) |
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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
−10.83700929014609229211346690775, −9.608392910602802350418992127649, −8.760156123609079454801693688144, −7.926848579810194372607515524147, −6.84257722433543424254256608417, −5.92340020067579220782283814091, −5.56647266488771618018473319632, −3.98134614250087262566711346080, −2.61052292141830878387349975518, −1.24374542182965726846331833186,
0.800536964207138214866170531866, 2.95798955990552239925869858484, 4.05718607296009922953587089035, 4.84368232129346976399746857508, 6.04947400416303218877906522520, 6.74552726448147471442422499723, 7.72369751350048662557839039247, 9.234319721505140041445591891651, 9.510124542903399770170959501977, 10.54754263855235102452103428872