L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.668 − 0.668i)3-s + 1.00i·4-s + (−2.17 + 0.508i)5-s + 0.945i·6-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s − 2.10i·9-s + (1.89 + 1.17i)10-s + (2.68 − 1.94i)11-s + (0.668 − 0.668i)12-s + (−1.94 + 1.94i)13-s − 1.00i·14-s + (1.79 + 1.11i)15-s − 1.00·16-s + (−3.61 − 3.61i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.386 − 0.386i)3-s + 0.500i·4-s + (−0.973 + 0.227i)5-s + 0.386i·6-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s − 0.701i·9-s + (0.600 + 0.373i)10-s + (0.809 − 0.586i)11-s + (0.193 − 0.193i)12-s + (−0.539 + 0.539i)13-s − 0.267i·14-s + (0.463 + 0.288i)15-s − 0.250·16-s + (−0.875 − 0.875i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0885458 + 0.120422i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0885458 + 0.120422i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.17 - 0.508i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (-2.68 + 1.94i)T \) |
good | 3 | \( 1 + (0.668 + 0.668i)T + 3iT^{2} \) |
| 13 | \( 1 + (1.94 - 1.94i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.61 + 3.61i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 23 | \( 1 + (-6.28 - 6.28i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.14T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + (6.35 - 6.35i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.80iT - 41T^{2} \) |
| 43 | \( 1 + (8.15 - 8.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.337 - 0.337i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.85 - 5.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 + 0.995iT - 61T^{2} \) |
| 67 | \( 1 + (2.69 - 2.69i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 + (-3.48 + 3.48i)T - 73iT^{2} \) |
| 79 | \( 1 + 8.68T + 79T^{2} \) |
| 83 | \( 1 + (8.86 - 8.86i)T - 83iT^{2} \) |
| 89 | \( 1 - 6.56iT - 89T^{2} \) |
| 97 | \( 1 + (5.10 - 5.10i)T - 97iT^{2} \) |
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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
−10.96266969657428837668147994214, −9.487939716212957801013800266203, −9.030779421697388634228320463363, −8.088499178891330667011729832902, −7.00097159233556919635751619896, −6.62393435090101045032215281505, −5.09879595222110631335569005328, −3.96001931280154486799154913419, −3.03127166031543486288024259522, −1.43738651633263997194741963116,
0.097041379751189051210382156895, 2.00149653736000791140015231824, 3.92108122109151629380194640604, 4.64511239253406231668691982149, 5.52239080796658705914909117533, 6.90964867579183561669938796356, 7.33965140261468590328888205536, 8.537089692870321043377401408949, 8.870928023022276113158341845652, 10.29354663397337889391288647105