L(s) = 1 | + (−0.656 + 1.88i)2-s − 1.73·3-s + (−3.13 − 2.48i)4-s + (−3.27 + 3.77i)5-s + (1.13 − 3.27i)6-s − 9.55·7-s + (6.74 − 4.29i)8-s + 2.99·9-s + (−4.98 − 8.66i)10-s + 9.92i·11-s + (5.43 + 4.29i)12-s − 7.55i·13-s + (6.27 − 18.0i)14-s + (5.67 − 6.54i)15-s + (3.68 + 15.5i)16-s + 17.1i·17-s + ⋯ |
L(s) = 1 | + (−0.328 + 0.944i)2-s − 0.577·3-s + (−0.784 − 0.620i)4-s + (−0.654 + 0.755i)5-s + (0.189 − 0.545i)6-s − 1.36·7-s + (0.843 − 0.537i)8-s + 0.333·9-s + (−0.498 − 0.866i)10-s + 0.902i·11-s + (0.452 + 0.358i)12-s − 0.581i·13-s + (0.448 − 1.28i)14-s + (0.378 − 0.436i)15-s + (0.230 + 0.973i)16-s + 1.01i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0326529 - 0.347257i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0326529 - 0.347257i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.656 - 1.88i)T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (3.27 - 3.77i)T \) |
good | 7 | \( 1 + 9.55T + 49T^{2} \) |
| 11 | \( 1 - 9.92iT - 121T^{2} \) |
| 13 | \( 1 + 7.55iT - 169T^{2} \) |
| 17 | \( 1 - 17.1iT - 289T^{2} \) |
| 19 | \( 1 - 26.1iT - 361T^{2} \) |
| 23 | \( 1 - 1.67T + 529T^{2} \) |
| 29 | \( 1 + 0.350T + 841T^{2} \) |
| 31 | \( 1 + 46.0iT - 961T^{2} \) |
| 37 | \( 1 - 22.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 77.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 22.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 94.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 29.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 7.19iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 46.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 24.1T + 6.88e3T^{2} \) |
| 89 | \( 1 - 100.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 131. iT - 9.40e3T^{2} \) |
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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
−15.45588776038028330318876310598, −14.81140742090365128787248536345, −13.22980647629621220905128105882, −12.21901717537085838442612121005, −10.47118775739243339705745046344, −9.807852378991962489616833439487, −8.006210312686846544642177998281, −6.85515723896523060687612931322, −5.90506249402014908128425721758, −3.91882163760871580695074459189,
0.39000399673529603641987553530, 3.36052077497821543269114841007, 4.96697171481084920974518127229, 6.95744753738029799004715698106, 8.713567162917910408356807817308, 9.585101844123470384228554636692, 11.02009982761722897831261773656, 11.90773595067373500035384761910, 12.85923756987819275939615498040, 13.72280329550463806890345526526