L(s) = 1 | + (−0.547 − 0.949i)2-s + (−1.68 − 2.91i)3-s + (0.399 − 0.691i)4-s + (−1.84 + 3.19i)6-s + (−0.795 + 1.37i)7-s − 3.06·8-s + (−4.16 + 7.20i)9-s + (−1.84 − 3.19i)11-s − 2.68·12-s + (3.21 − 1.63i)13-s + 1.74·14-s + (0.882 + 1.52i)16-s + (−1.49 + 2.58i)17-s + 9.11·18-s + (1.39 − 2.42i)19-s + ⋯ |
L(s) = 1 | + (−0.387 − 0.671i)2-s + (−0.971 − 1.68i)3-s + (0.199 − 0.345i)4-s + (−0.752 + 1.30i)6-s + (−0.300 + 0.520i)7-s − 1.08·8-s + (−1.38 + 2.40i)9-s + (−0.555 − 0.962i)11-s − 0.775·12-s + (0.891 − 0.453i)13-s + 0.466·14-s + (0.220 + 0.381i)16-s + (−0.361 + 0.626i)17-s + 2.14·18-s + (0.321 − 0.556i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.259719 + 0.322727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.259719 + 0.322727i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 + (-3.21 + 1.63i)T \) |
good | 2 | \( 1 + (0.547 + 0.949i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.68 + 2.91i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (0.795 - 1.37i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.84 + 3.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (1.49 - 2.58i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.39 + 2.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.335 + 0.581i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.37 + 2.38i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.43T + 31T^{2} \) |
| 37 | \( 1 + (0.479 + 0.829i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.42 + 2.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.44 + 4.23i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.83T + 47T^{2} \) |
| 53 | \( 1 + 2.70T + 53T^{2} \) |
| 59 | \( 1 + (-5.17 + 8.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.71 - 6.43i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.76 + 9.98i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.28 - 7.42i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 3.72T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + (0.826 + 1.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.12 - 1.95i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.07301302754863353634735324716, −10.54401821650306555100958009300, −8.955996774465326121344576338919, −8.112937238503238129273530333861, −6.90152285643176020934443795974, −5.92451738963955669441373586872, −5.57294638996977729968995568144, −2.92536223700126030693975628002, −1.73230534991772285646266140660, −0.36304622185617679769022233344,
3.30855400539417621871621452516, 4.28453488737234209050662763348, 5.43099376035393462081097303927, 6.40189669876616271690706526688, 7.36557956553132273570047350687, 8.766788680741436015635418004468, 9.522083296482535727136917589320, 10.31818802703695770858273103855, 11.22287276814048962237666425118, 11.91300881827255369169898783041