L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.781 + 2.52i)7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 4.90·11-s + (−0.707 − 0.707i)12-s + (3.41 − 3.41i)13-s + (1.23 + 2.33i)14-s − 1.00·16-s + (−2.74 − 2.74i)17-s + (−0.707 − 0.707i)18-s + 1.26·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.408 − 0.408i)3-s − 0.500i·4-s − 0.408i·6-s + (−0.295 + 0.955i)7-s + (−0.250 − 0.250i)8-s − 0.333i·9-s + 1.47·11-s + (−0.204 − 0.204i)12-s + (0.946 − 0.946i)13-s + (0.330 + 0.625i)14-s − 0.250·16-s + (−0.666 − 0.666i)17-s + (−0.166 − 0.166i)18-s + 0.290·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.547765723\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.547765723\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.781 - 2.52i)T \) |
good | 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.74 + 2.74i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + (-1.05 - 1.05i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.76iT - 29T^{2} \) |
| 31 | \( 1 + 7.05iT - 31T^{2} \) |
| 37 | \( 1 + (-4.74 + 4.74i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.44iT - 41T^{2} \) |
| 43 | \( 1 + (-7.58 - 7.58i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.734 + 0.734i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.26 + 1.26i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.39T + 59T^{2} \) |
| 61 | \( 1 + 2.29iT - 61T^{2} \) |
| 67 | \( 1 + (3.05 - 3.05i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + (5.04 - 5.04i)T - 73iT^{2} \) |
| 79 | \( 1 - 14.8iT - 79T^{2} \) |
| 83 | \( 1 + (-9.29 + 9.29i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + (-8.42 - 8.42i)T + 97iT^{2} \) |
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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
−9.419539805932777697472858785629, −9.190419608545910005060756700879, −8.232859298789142063132901461069, −7.08942417412527676957967642270, −6.19097024826436950689469366601, −5.56401799796586989752851069139, −4.23102288777934554175493012092, −3.31707238585123202112465468635, −2.39561772553377939398829243101, −1.10362220488048397693875906722,
1.49502798341098413116497708944, 3.19802917032753436142813698951, 4.11701679192317225718125102843, 4.46229730029075993078618710825, 6.06936130497904101453470853055, 6.61421359027074647245177876407, 7.42562744406473080972497134492, 8.560273687915781282301788929629, 9.073190318746114635457332572954, 9.984399856867989482852147694404