L(s) = 1 | + (−0.0703 + 0.0703i)3-s + (−2.16 + 0.574i)5-s + (−2.58 + 0.562i)7-s + 2.99i·9-s + 0.777·11-s + (−3.93 + 3.93i)13-s + (0.111 − 0.192i)15-s + (0.982 + 0.982i)17-s − 1.14·19-s + (0.142 − 0.221i)21-s + (−1.46 − 1.46i)23-s + (4.34 − 2.48i)25-s + (−0.421 − 0.421i)27-s + 4.69i·29-s − 6.45i·31-s + ⋯ |
L(s) = 1 | + (−0.0405 + 0.0405i)3-s + (−0.966 + 0.256i)5-s + (−0.977 + 0.212i)7-s + 0.996i·9-s + 0.234·11-s + (−1.09 + 1.09i)13-s + (0.0288 − 0.0496i)15-s + (0.238 + 0.238i)17-s − 0.263·19-s + (0.0310 − 0.0482i)21-s + (−0.304 − 0.304i)23-s + (0.868 − 0.496i)25-s + (−0.0810 − 0.0810i)27-s + 0.870i·29-s − 1.15i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.648 - 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.239629 + 0.519140i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.239629 + 0.519140i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.16 - 0.574i)T \) |
| 7 | \( 1 + (2.58 - 0.562i)T \) |
good | 3 | \( 1 + (0.0703 - 0.0703i)T - 3iT^{2} \) |
| 11 | \( 1 - 0.777T + 11T^{2} \) |
| 13 | \( 1 + (3.93 - 3.93i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.982 - 0.982i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 + (1.46 + 1.46i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.69iT - 29T^{2} \) |
| 31 | \( 1 + 6.45iT - 31T^{2} \) |
| 37 | \( 1 + (1.30 - 1.30i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.81iT - 41T^{2} \) |
| 43 | \( 1 + (7.13 + 7.13i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.08 + 2.08i)T + 53iT^{2} \) |
| 59 | \( 1 - 8.29T + 59T^{2} \) |
| 61 | \( 1 + 5.88iT - 61T^{2} \) |
| 67 | \( 1 + (6.30 - 6.30i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + (-7.23 + 7.23i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.83iT - 79T^{2} \) |
| 83 | \( 1 + (10.4 - 10.4i)T - 83iT^{2} \) |
| 89 | \( 1 - 3.41T + 89T^{2} \) |
| 97 | \( 1 + (-8.50 - 8.50i)T + 97iT^{2} \) |
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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
−12.14022125076558984591761276869, −11.35241440194788019603514880922, −10.33100690217455832114288118289, −9.425349658933037703967395062768, −8.285644756452227228379599245191, −7.29958430265785302346793932854, −6.45083218492130518716973199758, −4.94643648358127694125523093418, −3.85404939569987442786058649226, −2.45287489582426247986309839198,
0.42009032164349763740034057266, 3.06624223557928701608013514503, 4.00258851473475914398238046382, 5.42635424488939854036002922907, 6.72229991318820788455258472182, 7.51853949681672417058626194344, 8.666440384544602890468987793693, 9.659022797953064780877590810349, 10.47462456245219395991875371644, 11.83357851512005536640844407317