L(s) = 1 | + 1.99i·2-s + (1.04 − 1.04i)3-s − 1.97·4-s + 1.71i·5-s + (2.08 + 2.08i)6-s + (−0.707 + 0.707i)7-s + 0.0591i·8-s + 0.801i·9-s − 3.40·10-s + (2.84 − 2.84i)11-s + (−2.06 + 2.06i)12-s + (−0.592 + 0.592i)13-s + (−1.40 − 1.40i)14-s + (1.79 + 1.79i)15-s − 4.05·16-s + (4.76 + 4.76i)17-s + ⋯ |
L(s) = 1 | + 1.40i·2-s + (0.605 − 0.605i)3-s − 0.985·4-s + 0.764i·5-s + (0.852 + 0.852i)6-s + (−0.267 + 0.267i)7-s + 0.0209i·8-s + 0.267i·9-s − 1.07·10-s + (0.856 − 0.856i)11-s + (−0.596 + 0.596i)12-s + (−0.164 + 0.164i)13-s + (−0.376 − 0.376i)14-s + (0.462 + 0.462i)15-s − 1.01·16-s + (1.15 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.773369 + 1.30990i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.773369 + 1.30990i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.359 + 6.39i)T \) |
good | 2 | \( 1 - 1.99iT - 2T^{2} \) |
| 3 | \( 1 + (-1.04 + 1.04i)T - 3iT^{2} \) |
| 5 | \( 1 - 1.71iT - 5T^{2} \) |
| 11 | \( 1 + (-2.84 + 2.84i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.592 - 0.592i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.76 - 4.76i)T + 17iT^{2} \) |
| 19 | \( 1 + (1.80 + 1.80i)T + 19iT^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + (-6.01 + 6.01i)T - 29iT^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 8.10T + 37T^{2} \) |
| 43 | \( 1 + 10.4iT - 43T^{2} \) |
| 47 | \( 1 + (4.86 + 4.86i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.81 + 2.81i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 3.67iT - 61T^{2} \) |
| 67 | \( 1 + (-5.59 - 5.59i)T + 67iT^{2} \) |
| 71 | \( 1 + (-8.63 + 8.63i)T - 71iT^{2} \) |
| 73 | \( 1 - 8.12iT - 73T^{2} \) |
| 79 | \( 1 + (9.11 - 9.11i)T - 79iT^{2} \) |
| 83 | \( 1 - 0.0385T + 83T^{2} \) |
| 89 | \( 1 + (-5.40 + 5.40i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.41 + 3.41i)T + 97iT^{2} \) |
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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
−12.31145410983242294101686056425, −11.20020705812729200300061024774, −10.06301959989148011920637710396, −8.695930401342294831362803930409, −8.162624734324752422636632460864, −7.20025011653110349389316559599, −6.39167752773032983600861682630, −5.60995130740814439613970423273, −3.83239207982754543715473218076, −2.29674016890426737113300574957,
1.24290263381542299964742459032, 2.89997559429701192599900524938, 3.92524086289822521056031573928, 4.76305011995321227808365447721, 6.55218186582390148805338072710, 7.993794634031850447095911045218, 9.344029156160160810172766677329, 9.553711216360355282056400921317, 10.39477341216990634798477375866, 11.66333079936714697799162431419