L(s) = 1 | + (−1.38 + 0.296i)2-s + (−1.71 − 0.246i)3-s + (1.82 − 0.820i)4-s + 2.85i·5-s + (2.44 − 0.168i)6-s + 2.12i·7-s + (−2.27 + 1.67i)8-s + (2.87 + 0.843i)9-s + (−0.846 − 3.94i)10-s + 5.05·11-s + (−3.32 + 0.958i)12-s + 1.45·13-s + (−0.630 − 2.93i)14-s + (0.701 − 4.89i)15-s + (2.65 − 2.99i)16-s + 0.803i·17-s + ⋯ |
L(s) = 1 | + (−0.977 + 0.209i)2-s + (−0.989 − 0.142i)3-s + (0.911 − 0.410i)4-s + 1.27i·5-s + (0.997 − 0.0688i)6-s + 0.802i·7-s + (−0.805 + 0.592i)8-s + (0.959 + 0.281i)9-s + (−0.267 − 1.24i)10-s + 1.52·11-s + (−0.960 + 0.276i)12-s + 0.403·13-s + (−0.168 − 0.784i)14-s + (0.181 − 1.26i)15-s + (0.663 − 0.748i)16-s + 0.194i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.464038 + 0.616539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.464038 + 0.616539i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 - 0.296i)T \) |
| 3 | \( 1 + (1.71 + 0.246i)T \) |
| 67 | \( 1 + iT \) |
good | 5 | \( 1 - 2.85iT - 5T^{2} \) |
| 7 | \( 1 - 2.12iT - 7T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 - 0.803iT - 17T^{2} \) |
| 19 | \( 1 - 3.66iT - 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 8.38iT - 29T^{2} \) |
| 31 | \( 1 + 2.39iT - 31T^{2} \) |
| 37 | \( 1 - 1.53T + 37T^{2} \) |
| 41 | \( 1 - 0.167iT - 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 4.56T + 59T^{2} \) |
| 61 | \( 1 + 7.45T + 61T^{2} \) |
| 71 | \( 1 + 15.5T + 71T^{2} \) |
| 73 | \( 1 - 0.104T + 73T^{2} \) |
| 79 | \( 1 - 6.42iT - 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 + 6.70iT - 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−10.51561843042299437451416869974, −9.743308919591234913482550137385, −8.954758110178129316993137885233, −7.78633735714630045055463868478, −6.99602554748467530385679866697, −6.06777055669600585444879977980, −5.97059138777998959408721750386, −4.12573437342788068098377341473, −2.67507057079669281330504990904, −1.36124798913679607621453340219,
0.72968328434790877161942234966, 1.45256085679136644710067946768, 3.64959728672534208159817896743, 4.56108238240381300325055068119, 5.65822841108021421354237238232, 6.82227878426309416081134028761, 7.23450384603978982578221504742, 8.827972162000425265565554759836, 8.956487172934158491820764543482, 10.03294647156070803670336434388