| L(s) = 1 | + (−0.415 + 0.909i)2-s − 3-s + (−0.654 − 0.755i)4-s + (1.86 + 0.548i)5-s + (0.415 − 0.909i)6-s + (−0.239 − 1.66i)7-s + (0.959 − 0.281i)8-s + 9-s + (−1.27 + 1.47i)10-s + (−0.826 − 3.21i)11-s + (0.654 + 0.755i)12-s + (0.884 + 1.02i)13-s + (1.61 + 0.473i)14-s + (−1.86 − 0.548i)15-s + (−0.142 + 0.989i)16-s + (−4.88 − 3.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s − 0.577·3-s + (−0.327 − 0.377i)4-s + (0.834 + 0.245i)5-s + (0.169 − 0.371i)6-s + (−0.0904 − 0.629i)7-s + (0.339 − 0.0996i)8-s + 0.333·9-s + (−0.402 + 0.464i)10-s + (−0.249 − 0.968i)11-s + (0.189 + 0.218i)12-s + (0.245 + 0.283i)13-s + (0.431 + 0.126i)14-s + (−0.481 − 0.141i)15-s + (−0.0355 + 0.247i)16-s + (−1.18 − 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.866508 - 0.363448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.866508 - 0.363448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + (0.826 + 3.21i)T \) |
| good | 5 | \( 1 + (-1.86 - 0.548i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.239 + 1.66i)T + (-6.71 + 1.97i)T^{2} \) |
| 13 | \( 1 + (-0.884 - 1.02i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.88 + 3.14i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.535 - 0.344i)T + (7.89 - 17.2i)T^{2} \) |
| 23 | \( 1 + (-0.750 + 5.22i)T + (-22.0 - 6.47i)T^{2} \) |
| 29 | \( 1 + (2.77 - 1.78i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.17 + 5.97i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (1.95 - 2.25i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-1.37 + 3.01i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.20 + 2.11i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (2.01 + 4.41i)T + (-30.7 + 35.5i)T^{2} \) |
| 53 | \( 1 + (-0.333 - 2.32i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-1.54 - 3.38i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 2.92i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-6.22 + 13.6i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.720 + 0.463i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.722 + 5.02i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (0.183 + 0.0539i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.317 + 2.20i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-5.28 - 3.39i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.70 + 2.26i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.36805855962098537779479497795, −9.392344157091323226486612810048, −8.632630621784714781619764966361, −7.53091681009184273295600862302, −6.57463464085061734283086384602, −6.10293763257478004336589606456, −5.07619019499487102262471882976, −4.02592557702375531881194750906, −2.34954499464920836755043343482, −0.58766796694592424785884235379,
1.54430195827358393403450259001, 2.51192388609061111092067818469, 4.06595941669830678151491930556, 5.13095862626829752292438076767, 5.93516989833321833990454353391, 6.94690473612587858915501008769, 8.101654248744452829309810250742, 9.106629014717190212759257018976, 9.686164846457941431313992827739, 10.49715023942243039071388211836
