L(s) = 1 | + (0.576 − 0.333i)2-s + (1.24 + 2.15i)3-s + (−0.778 + 1.34i)4-s + (1.43 + 0.826i)6-s + (0.509 + 0.293i)7-s + 2.36i·8-s + (−1.58 + 2.74i)9-s + (−2.58 + 1.49i)11-s − 3.86·12-s + (−1.93 − 3.04i)13-s + 0.391·14-s + (−0.767 − 1.32i)16-s + (3.28 − 5.69i)17-s + 2.10i·18-s + (5.19 + 3.00i)19-s + ⋯ |
L(s) = 1 | + (0.407 − 0.235i)2-s + (0.716 + 1.24i)3-s + (−0.389 + 0.673i)4-s + (0.584 + 0.337i)6-s + (0.192 + 0.111i)7-s + 0.837i·8-s + (−0.527 + 0.913i)9-s + (−0.779 + 0.450i)11-s − 1.11·12-s + (−0.537 − 0.843i)13-s + 0.104·14-s + (−0.191 − 0.332i)16-s + (0.797 − 1.38i)17-s + 0.496i·18-s + (1.19 + 0.688i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23855 + 1.27740i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23855 + 1.27740i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (1.93 + 3.04i)T \) |
good | 2 | \( 1 + (-0.576 + 0.333i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.24 - 2.15i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-0.509 - 0.293i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.58 - 1.49i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.28 + 5.69i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.19 - 3.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.335 - 0.580i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.94 - 6.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.53iT - 31T^{2} \) |
| 37 | \( 1 + (-6.70 + 3.87i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.629 - 0.363i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.503 - 0.871i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.15iT - 47T^{2} \) |
| 53 | \( 1 - 4.94T + 53T^{2} \) |
| 59 | \( 1 + (12.3 + 7.13i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.03 + 6.98i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.551 - 0.318i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.79 + 4.50i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 16.8iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 0.370iT - 83T^{2} \) |
| 89 | \( 1 + (8.00 - 4.61i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.99 + 3.46i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.00163354858605316057401591437, −10.79189254470658261506138842327, −9.863668068311630156028305482420, −9.235379358963058380318595613563, −8.102312959273602975912125810764, −7.47006987576592602232843836025, −5.20056777795459939875783911704, −4.85525295171190808436905588720, −3.39638628088924015855668501698, −2.84333527637704160232273294602,
1.18745379162823854317512857758, 2.67108408394628950730925545115, 4.27376828093807937627347183358, 5.57477379520356922215043140278, 6.51927950745228286596494842919, 7.55219030908060388058194221737, 8.297970030173016266872837168543, 9.429035527544014024326145279200, 10.33452454113281046062276924177, 11.60955578980813973137119610581