| L(s) = 1 | + (−1.48 − 0.0171i)2-s + (−0.358 + 1.56i)3-s + (0.212 + 0.00489i)4-s + (1.55 − 2.92i)5-s + (0.559 − 2.32i)6-s + (−0.286 − 0.705i)7-s + (2.65 + 0.0917i)8-s + (0.370 + 0.178i)9-s + (−2.36 + 4.32i)10-s + (−1.81 − 1.88i)11-s + (−0.0837 + 0.331i)12-s + (−3.95 − 0.596i)13-s + (0.413 + 1.05i)14-s + (4.02 + 3.48i)15-s + (−4.37 − 0.201i)16-s + (0.346 − 2.05i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.0121i)2-s + (−0.206 + 0.905i)3-s + (0.106 + 0.00244i)4-s + (0.696 − 1.30i)5-s + (0.228 − 0.949i)6-s + (−0.108 − 0.266i)7-s + (0.939 + 0.0324i)8-s + (0.123 + 0.0594i)9-s + (−0.747 + 1.36i)10-s + (−0.545 − 0.568i)11-s + (−0.0241 + 0.0957i)12-s + (−1.09 − 0.165i)13-s + (0.110 + 0.281i)14-s + (1.04 + 0.900i)15-s + (−1.09 − 0.0503i)16-s + (0.0839 − 0.498i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.261443 - 0.361880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261443 - 0.361880i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + (-8.14 + 21.9i)T \) |
| good | 2 | \( 1 + (1.48 + 0.0171i)T + (1.99 + 0.0460i)T^{2} \) |
| 3 | \( 1 + (0.358 - 1.56i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (-1.55 + 2.92i)T + (-2.79 - 4.14i)T^{2} \) |
| 7 | \( 1 + (0.286 + 0.705i)T + (-5.02 + 4.87i)T^{2} \) |
| 11 | \( 1 + (1.81 + 1.88i)T + (-0.442 + 10.9i)T^{2} \) |
| 13 | \( 1 + (3.95 + 0.596i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-0.346 + 2.05i)T + (-16.0 - 5.56i)T^{2} \) |
| 19 | \( 1 + (2.33 - 1.25i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (3.53 + 0.828i)T + (20.6 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-3.74 - 2.13i)T + (14.7 + 24.9i)T^{2} \) |
| 31 | \( 1 + (7.64 + 2.75i)T + (23.8 + 19.7i)T^{2} \) |
| 37 | \( 1 + (-6.18 + 4.98i)T + (7.81 - 36.1i)T^{2} \) |
| 41 | \( 1 + (3.86 + 6.68i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.607 + 6.19i)T + (-42.1 - 8.35i)T^{2} \) |
| 47 | \( 1 + (-0.124 + 0.0100i)T + (46.3 - 7.53i)T^{2} \) |
| 53 | \( 1 + (8.57 - 2.97i)T + (41.6 - 32.8i)T^{2} \) |
| 59 | \( 1 + (-8.53 + 3.63i)T + (40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 1.32i)T + (39.1 - 46.8i)T^{2} \) |
| 67 | \( 1 + (8.79 + 8.54i)T + (1.92 + 66.9i)T^{2} \) |
| 71 | \( 1 + (2.91 - 0.576i)T + (65.6 - 27.0i)T^{2} \) |
| 73 | \( 1 + (0.285 - 0.246i)T + (10.4 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.136 - 0.0978i)T + (25.4 + 74.7i)T^{2} \) |
| 83 | \( 1 + (5.01 + 3.16i)T + (35.5 + 74.9i)T^{2} \) |
| 89 | \( 1 + (-0.318 - 3.68i)T + (-87.6 + 15.2i)T^{2} \) |
| 97 | \( 1 + (-1.39 - 10.4i)T + (-93.6 + 25.3i)T^{2} \) |
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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
−10.27597356098563915018070480846, −9.584979836926153550710118259150, −9.061069657401485631129571476177, −8.164216094938619104497457475896, −7.26028715709606988343743384997, −5.57577254160996137797150878087, −4.94772370614124510109691916304, −4.05174991118053101040739333768, −1.99386297232026418936112525895, −0.36887827269710792157675890974,
1.71352740818050573655860141446, 2.63661434237814631703390532669, 4.50323459919700249032384976563, 5.96343736681273591974126956603, 6.85474040328884068633600501316, 7.41784156883686542976126637535, 8.259929729226395659685951059325, 9.624278151129559602317750157396, 9.968886826211841595735401714455, 10.76086037478051564767078432735
