L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)20-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.309 + 0.951i)8-s + (−0.809 − 0.587i)9-s + 10-s − 12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (−0.809 + 0.587i)16-s + (−0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + (0.309 − 0.951i)19-s + (0.809 + 0.587i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9913238948 + 0.9697650696i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9913238948 + 0.9697650696i\) |
\(L(1)\) |
\(\approx\) |
\(1.219732698 + 0.7668342266i\) |
\(L(1)\) |
\(\approx\) |
\(1.219732698 + 0.7668342266i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.809 - 0.587i)T \) |
| 17 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 + 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.92695546899438077064516923007, −29.7426858177209147836021280841, −29.29170026490570206840740213516, −28.38698940291412384727711598174, −26.726297590666911171785178823565, −25.109326450604602302174206511031, −24.511257792990138944465729069040, −23.19868191333155194627159534636, −22.39278419963948652206120439105, −21.40685800606579416463694904291, −20.046160109936437452197342644033, −18.9049168119050562144389617683, −18.08568226612136036266897431267, −16.70229671702335172761754867314, −14.82050985807158446935500661687, −13.91232937686748418327031838648, −13.00232152460253927742130275969, −11.822398727821472462133223077453, −10.784753992449663968317824700728, −9.41519698387563615793830359095, −7.19006823790203154020251729714, −6.2305635075433427328844873371, −4.99191069220013488410129458335, −2.872941676570042368827334925973, −1.72612569056363330400636736493,
2.790752711064700332423271599631, 4.521477273188341623907474995450, 5.32346352798603951106336348544, 6.568733301320764713572357001242, 8.45265704371212955870829398169, 9.67439512799399461609663540920, 11.151838054458632646017193348708, 12.538670914303729435025340668448, 13.63945887374512705613801802759, 14.94295093466554678734851101208, 15.79921011650420131647259377732, 17.13170175417702845115483959496, 17.49929685517634714315747158387, 20.00006133741696880544804111423, 21.00366111877825470547864447171, 21.8617635731850582039917495496, 22.633814781454633199673892886253, 24.00021625516034222368237942798, 24.965832157281700295240287491394, 26.06643672350573742276215413713, 27.02409833093482319708181931799, 28.46188734507754438825342822638, 29.33640421689870373374637621838, 30.69168036418032878249535968486, 32.03748926478594271955658237702