L(s) = 1 | + (−1.38 + 0.282i)2-s + (2.74 + 0.735i)3-s + (1.84 − 0.782i)4-s + (1.86 − 1.23i)5-s + (−4.00 − 0.244i)6-s + (−0.517 + 2.59i)7-s + (−2.33 + 1.60i)8-s + (4.38 + 2.53i)9-s + (−2.23 + 2.23i)10-s + (1.41 + 2.44i)11-s + (5.62 − 0.792i)12-s + (−4.50 − 4.50i)13-s + (−0.0146 − 3.74i)14-s + (6.02 − 2.01i)15-s + (2.77 − 2.87i)16-s + (0.0288 − 0.107i)17-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.199i)2-s + (1.58 + 0.424i)3-s + (0.920 − 0.391i)4-s + (0.834 − 0.551i)5-s + (−1.63 − 0.0998i)6-s + (−0.195 + 0.980i)7-s + (−0.823 + 0.566i)8-s + (1.46 + 0.844i)9-s + (−0.707 + 0.707i)10-s + (0.425 + 0.737i)11-s + (1.62 − 0.228i)12-s + (−1.24 − 1.24i)13-s + (−0.00390 − 0.999i)14-s + (1.55 − 0.519i)15-s + (0.694 − 0.719i)16-s + (0.00700 − 0.0261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.847 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43110 + 0.411408i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43110 + 0.411408i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.38 - 0.282i)T \) |
| 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 7 | \( 1 + (0.517 - 2.59i)T \) |
good | 3 | \( 1 + (-2.74 - 0.735i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.50 + 4.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.0288 + 0.107i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.58 + 2.07i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.50 - 1.20i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + (-2.21 + 1.27i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.282 + 1.05i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 + (-3.01 + 3.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.16 - 8.09i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.53 - 5.73i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-4.70 + 2.71i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.201 - 0.116i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.423 - 1.58i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 0.772iT - 71T^{2} \) |
| 73 | \( 1 + (12.4 + 3.32i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.36 + 9.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.37 + 2.37i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.48 + 4.32i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.14 - 3.14i)T + 97iT^{2} \) |
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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
−12.09411321448771999001767087166, −10.25435896060643742123846701126, −9.834684867526526344717515647001, −9.049152911778937021324599544965, −8.428691188104660153841732190024, −7.50379501440393677613690179361, −6.11475902742923259070220394598, −4.80623471510313032286284744367, −2.80316675390644180450479229547, −2.08075237774716606692625290410,
1.73070325878522764606344380352, 2.73133776517396731509783108601, 3.93122456578579153658706366731, 6.51022065534334446926392369606, 7.02253000487274048178942257712, 8.095868764653406246047415067228, 8.917444072719660310317754036625, 9.841147894642329661781702760163, 10.31642733474246852370939123773, 11.67449872291196384875736707001