L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)3-s + 1.00i·4-s + (−2.10 + 0.741i)5-s + 1.00i·6-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (2.01 + 0.967i)10-s + 1.16·11-s + (0.707 − 0.707i)12-s + (−3.61 − 3.61i)13-s + (2.01 + 0.967i)15-s − 1.00·16-s + (4.48 − 4.48i)17-s + (0.707 − 0.707i)18-s − 2.15·19-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (−0.943 + 0.331i)5-s + 0.408i·6-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (0.637 + 0.305i)10-s + 0.352·11-s + (0.204 − 0.204i)12-s + (−1.00 − 1.00i)13-s + (0.520 + 0.249i)15-s − 0.250·16-s + (1.08 − 1.08i)17-s + (0.166 − 0.166i)18-s − 0.494·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3178308874\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3178308874\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (2.10 - 0.741i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.16T + 11T^{2} \) |
| 13 | \( 1 + (3.61 + 3.61i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.48 + 4.48i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 + (1.80 - 1.80i)T - 23iT^{2} \) |
| 29 | \( 1 - 7.29iT - 29T^{2} \) |
| 31 | \( 1 + 3.14iT - 31T^{2} \) |
| 37 | \( 1 + (2.88 + 2.88i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.21iT - 41T^{2} \) |
| 43 | \( 1 + (-2.87 + 2.87i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.84 - 7.84i)T - 47iT^{2} \) |
| 53 | \( 1 + (7.17 - 7.17i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.85T + 59T^{2} \) |
| 61 | \( 1 - 10.0iT - 61T^{2} \) |
| 67 | \( 1 + (-10.3 - 10.3i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 + (3.69 + 3.69i)T + 73iT^{2} \) |
| 79 | \( 1 + 0.468iT - 79T^{2} \) |
| 83 | \( 1 + (-9.87 - 9.87i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.12T + 89T^{2} \) |
| 97 | \( 1 + (-6.30 + 6.30i)T - 97iT^{2} \) |
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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
−9.801945473956604295596138020801, −8.894183128306567267161294305095, −7.78853692202852054316702702744, −7.57317854348727049943188907032, −6.68705008104554789997870479171, −5.50068408896293028233249939955, −4.58264946192388947319171804261, −3.40763314740498856191357695511, −2.63064254746715797768476109125, −1.06826749352163522023034670040,
0.18979948993923302925246470038, 1.79002430893774364614199623360, 3.52504968752672673796057337827, 4.39319872769365371435008885236, 5.11364356309204236578355737869, 6.22711455999357205927978503267, 6.89760538303167903878872607370, 7.88685490266691678303605926673, 8.403846221701883971475002594101, 9.362362707201684508479716543353