L(s) = 1 | + (0.618 + 1.61i)3-s − 2.23·5-s + (−1 − i)7-s + (−2.23 + 2.00i)9-s − 4.47i·11-s + (3 − 3i)13-s + (−1.38 − 3.61i)15-s + (2.23 − 2.23i)17-s + 2i·19-s + (1 − 2.23i)21-s + (−2.23 − 2.23i)23-s + 5.00·25-s + (−4.61 − 2.38i)27-s + 4.47·29-s + 4·31-s + ⋯ |
L(s) = 1 | + (0.356 + 0.934i)3-s − 0.999·5-s + (−0.377 − 0.377i)7-s + (−0.745 + 0.666i)9-s − 1.34i·11-s + (0.832 − 0.832i)13-s + (−0.356 − 0.934i)15-s + (0.542 − 0.542i)17-s + 0.458i·19-s + (0.218 − 0.487i)21-s + (−0.466 − 0.466i)23-s + 1.00·25-s + (−0.888 − 0.458i)27-s + 0.830·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14072 - 0.385863i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14072 - 0.385863i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 - 1.61i)T \) |
| 5 | \( 1 + 2.23T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.47iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + (2.23 + 2.23i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.94iT - 41T^{2} \) |
| 43 | \( 1 + (-3 + 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.70 - 6.70i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.23 + 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + (-1 - i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (6.70 + 6.70i)T + 83iT^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + (-9 - 9i)T + 97iT^{2} \) |
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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
−10.12044790747949962085215358384, −8.983490714114111136648138917787, −8.270987987101726918871656281561, −7.79504653210529528359344458657, −6.42680885105880373850851970383, −5.51813783879856691579658446804, −4.42170407726469852992939011993, −3.49147796184537358946725204729, −3.02065628922490800224158870246, −0.59507906019160650978863238508,
1.34231332985726759130732995103, 2.63551777097133147067110496024, 3.72827001220566881274346905860, 4.69698351076062491034953450516, 6.14788747801314144587627277009, 6.77957754034941056612945805805, 7.65308372489570943359819392403, 8.280765059514846230449735601012, 9.139560119166136652600021240673, 9.957997795679852682282419847126