L(s) = 1 | + (0.707 + 0.707i)2-s + (2.23 + 2.23i)3-s + 1.00i·4-s + (0.288 − 2.21i)5-s + 3.15i·6-s + (−1.64 + 2.06i)7-s + (−0.707 + 0.707i)8-s + 6.95i·9-s + (1.77 − 1.36i)10-s − 11-s + (−2.23 + 2.23i)12-s + (2.34 + 2.34i)13-s + (−2.62 + 0.298i)14-s + (5.59 − 4.30i)15-s − 1.00·16-s + (3.89 − 3.89i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (1.28 + 1.28i)3-s + 0.500i·4-s + (0.128 − 0.991i)5-s + 1.28i·6-s + (−0.622 + 0.782i)7-s + (−0.250 + 0.250i)8-s + 2.31i·9-s + (0.560 − 0.431i)10-s − 0.301·11-s + (−0.644 + 0.644i)12-s + (0.650 + 0.650i)13-s + (−0.702 + 0.0797i)14-s + (1.44 − 1.11i)15-s − 0.250·16-s + (0.944 − 0.944i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41234 + 2.48558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41234 + 2.48558i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.288 + 2.21i)T \) |
| 7 | \( 1 + (1.64 - 2.06i)T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + (-2.23 - 2.23i)T + 3iT^{2} \) |
| 13 | \( 1 + (-2.34 - 2.34i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.89 + 3.89i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.89T + 19T^{2} \) |
| 23 | \( 1 + (0.0242 - 0.0242i)T - 23iT^{2} \) |
| 29 | \( 1 - 8.09iT - 29T^{2} \) |
| 31 | \( 1 + 8.61iT - 31T^{2} \) |
| 37 | \( 1 + (-6.66 - 6.66i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.7iT - 41T^{2} \) |
| 43 | \( 1 + (1.62 - 1.62i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.09 + 5.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.27 + 6.27i)T - 53iT^{2} \) |
| 59 | \( 1 - 0.608T + 59T^{2} \) |
| 61 | \( 1 - 10.2iT - 61T^{2} \) |
| 67 | \( 1 + (5.29 + 5.29i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.98T + 71T^{2} \) |
| 73 | \( 1 + (-2.98 - 2.98i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.09iT - 79T^{2} \) |
| 83 | \( 1 + (4.36 + 4.36i)T + 83iT^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 + (-6.57 + 6.57i)T - 97iT^{2} \) |
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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
−10.21614558793787102845370805900, −9.486940407226916304023644022913, −8.857873752058643939746288738903, −8.388745386088436858716064282928, −7.33100058889531470538289615187, −5.84501495105825669360451806442, −5.10818556118085132542268798271, −4.20823237603808212998628771922, −3.35528174480988604481503784608, −2.30291750722612480638081983280,
1.16329395109999196179429616882, 2.47298251697771898375609672454, 3.24201713635951126265447495061, 3.93772071505747313027481110563, 6.01856656607811313353088773628, 6.48278031532106915775680089946, 7.56592901648993472833788129359, 8.016989129433756475856240487625, 9.229457371679078064181745315497, 10.16789739800580323105349065916