L(s) = 1 | + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (2.53 − 3.48i)5-s + (0.951 + 0.309i)7-s + (−0.309 − 0.951i)8-s − 4.31i·10-s + (−2.59 + 2.06i)11-s + (−3.41 − 4.69i)13-s + (0.951 − 0.309i)14-s + (−0.809 − 0.587i)16-s + (3.36 + 2.44i)17-s + (2.74 − 0.890i)19-s + (−2.53 − 3.48i)20-s + (−0.891 + 3.19i)22-s − 5.92i·23-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (1.13 − 1.55i)5-s + (0.359 + 0.116i)7-s + (−0.109 − 0.336i)8-s − 1.36i·10-s + (−0.783 + 0.621i)11-s + (−0.946 − 1.30i)13-s + (0.254 − 0.0825i)14-s + (−0.202 − 0.146i)16-s + (0.815 + 0.592i)17-s + (0.628 − 0.204i)19-s + (−0.566 − 0.779i)20-s + (−0.190 + 0.681i)22-s − 1.23i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.664001810\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.664001810\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 + (2.59 - 2.06i)T \) |
good | 5 | \( 1 + (-2.53 + 3.48i)T + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (3.41 + 4.69i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.36 - 2.44i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.74 + 0.890i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 5.92iT - 23T^{2} \) |
| 29 | \( 1 + (2.53 - 7.79i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.02 + 3.64i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.74 - 8.43i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.31 - 4.03i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 2.81iT - 43T^{2} \) |
| 47 | \( 1 + (3.74 - 1.21i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.28 + 7.27i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-6.92 - 2.25i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.553 + 0.762i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + (0.423 - 0.583i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.71 + 1.85i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-8.25 - 11.3i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.38 - 3.18i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 4.27iT - 89T^{2} \) |
| 97 | \( 1 + (0.399 - 0.290i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.681824327033003009081261253559, −8.426831880121790820952789713055, −7.966588735485730598302193780133, −6.62819914529451631694473112214, −5.38157907762686649774444142959, −5.27388864321639990611194034357, −4.50279204635413481875882145513, −2.95533332135039462995735138841, −1.99515688899690059393538978912, −0.889712007630160244523618186905,
1.98496469686018486322574889887, 2.80129214112550343058952120800, 3.71005398808492782577254381962, 5.10749773407808500571598950283, 5.70005766338117819583160165362, 6.53047613675164054467794266531, 7.34588030734590467889374939973, 7.77466734723764475142007609133, 9.283107415855681987756499402953, 9.811935946557355928607186982986