| L(s) = 1 | + (−4.89 − 2.82i)5-s + (−1.22 + 0.707i)11-s + 16·13-s + (−14.6 + 8.48i)17-s + (4 − 6.92i)19-s + (−11.0 − 6.36i)23-s + (3.49 + 6.06i)25-s + 24.0i·29-s + (−28 − 48.4i)31-s + (−12 + 20.7i)37-s − 50.9i·41-s + 40·43-s + (9.79 + 5.65i)47-s + (−37.9 + 21.9i)53-s + 8·55-s + ⋯ |
| L(s) = 1 | + (−0.979 − 0.565i)5-s + (−0.111 + 0.0642i)11-s + 1.23·13-s + (−0.864 + 0.499i)17-s + (0.210 − 0.364i)19-s + (−0.479 − 0.276i)23-s + (0.139 + 0.242i)25-s + 0.829i·29-s + (−0.903 − 1.56i)31-s + (−0.324 + 0.561i)37-s − 1.24i·41-s + 0.930·43-s + (0.208 + 0.120i)47-s + (−0.716 + 0.413i)53-s + 0.145·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5739928739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5739928739\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (4.89 + 2.82i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (1.22 - 0.707i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 16T + 169T^{2} \) |
| 17 | \( 1 + (14.6 - 8.48i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4 + 6.92i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.0 + 6.36i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 24.0iT - 841T^{2} \) |
| 31 | \( 1 + (28 + 48.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (12 - 20.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-9.79 - 5.65i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (37.9 - 21.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (9.79 - 5.65i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-20 + 34.6i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-13 - 22.5i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-44 - 76.2i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (41 - 71.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (53.8 + 31.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 40T + 9.40e3T^{2} \) |
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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
−9.092104270833421299227213782769, −8.562246301384353833017437347088, −7.903789735190473145150245605716, −7.04692002391294238262584802144, −6.13879739317829638120090574035, −5.25368168355225188034293074451, −4.13596495562413011793395641890, −3.78710255227837975351459205406, −2.36567861236719461821320432931, −1.04116420022326075217796287657,
0.17481715188484436300445327000, 1.68091056371612803703333520579, 3.06046002525419224696499401066, 3.73508597490068532879058277386, 4.59862128261504098681545215115, 5.74152996355106972236508188361, 6.55102482696017744211687823477, 7.35975553821655303268953265953, 8.028172090715951667839770872267, 8.798937689129634001575865417450
