L(s) = 1 | + (−0.344 − 0.288i)2-s + (−1.83 + 0.669i)3-s + (−0.312 − 1.77i)4-s + (0.826 + 0.300i)6-s + (1.03 + 1.78i)7-s + (−0.853 + 1.47i)8-s + (0.635 − 0.533i)9-s + (1.15 − 1.99i)11-s + (1.75 + 3.04i)12-s + (4.13 + 1.50i)13-s + (0.160 − 0.911i)14-s + (−2.65 + 0.967i)16-s + (−4.19 − 3.51i)17-s − 0.372·18-s + (3.07 − 3.09i)19-s + ⋯ |
L(s) = 1 | + (−0.243 − 0.204i)2-s + (−1.06 + 0.386i)3-s + (−0.156 − 0.885i)4-s + (0.337 + 0.122i)6-s + (0.389 + 0.674i)7-s + (−0.301 + 0.522i)8-s + (0.211 − 0.177i)9-s + (0.347 − 0.602i)11-s + (0.507 + 0.879i)12-s + (1.14 + 0.417i)13-s + (0.0429 − 0.243i)14-s + (−0.664 + 0.241i)16-s + (−1.01 − 0.853i)17-s − 0.0878·18-s + (0.704 − 0.709i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.606765 - 0.456822i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.606765 - 0.456822i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-3.07 + 3.09i)T \) |
good | 2 | \( 1 + (0.344 + 0.288i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (1.83 - 0.669i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-1.03 - 1.78i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.15 + 1.99i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.13 - 1.50i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (4.19 + 3.51i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (1.01 + 5.72i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.21 + 3.53i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.378 + 0.656i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.22T + 37T^{2} \) |
| 41 | \( 1 + (-6.14 + 2.23i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.549 + 3.11i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.87 + 4.08i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.668 + 3.79i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.95 - 1.63i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.72 - 9.78i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.30 + 1.09i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.71 + 9.70i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (9.60 - 3.49i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (2.26 - 0.824i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.53 - 6.11i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.19 - 0.798i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-5.22 - 4.38i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.96550332022955609284154650429, −10.16253276382365336798581173074, −8.972188963310063790374741758525, −8.606459623666898057172157944895, −6.75969985399285775059771683857, −5.98278506392127974993109647424, −5.23670787933352794634216691757, −4.33738280633258473167471985873, −2.39605398777607861854700997709, −0.66613859807107060248989181727,
1.30415107745261657625229987571, 3.45478315516971410185468031726, 4.42760621923524727577865268615, 5.76027228111648031829859429194, 6.65737383371943818006181820077, 7.46413782300757510619173734813, 8.342313419307862863316021445847, 9.292145148261760807679179693527, 10.55573634439701785636168206318, 11.23157014351352060131991444397