L(s) = 1 | + (−27.2 + 27.2i)2-s + (84.7 + 111. i)3-s − 974. i·4-s + (−5.35e3 − 739. i)6-s + (4.32e3 + 4.32e3i)7-s + (1.26e4 + 1.26e4i)8-s + (−5.33e3 + 1.89e4i)9-s − 9.19e4i·11-s + (1.08e5 − 8.25e4i)12-s + (9.65e3 − 9.65e3i)13-s − 2.35e5·14-s − 1.88e5·16-s + (2.37e5 − 2.37e5i)17-s + (−3.71e5 − 6.61e5i)18-s + 8.14e5i·19-s + ⋯ |
L(s) = 1 | + (−1.20 + 1.20i)2-s + (0.603 + 0.797i)3-s − 1.90i·4-s + (−1.68 − 0.232i)6-s + (0.681 + 0.681i)7-s + (1.08 + 1.08i)8-s + (−0.270 + 0.962i)9-s − 1.89i·11-s + (1.51 − 1.14i)12-s + (0.0937 − 0.0937i)13-s − 1.64·14-s − 0.718·16-s + (0.688 − 0.688i)17-s + (−0.833 − 1.48i)18-s + 1.43i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.325323 + 1.35973i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.325323 + 1.35973i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-84.7 - 111. i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (27.2 - 27.2i)T - 512iT^{2} \) |
| 7 | \( 1 + (-4.32e3 - 4.32e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 + 9.19e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-9.65e3 + 9.65e3i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (-2.37e5 + 2.37e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 8.14e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (-6.20e5 - 6.20e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 - 6.40e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.64e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + (-1.23e7 - 1.23e7i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 - 1.60e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (1.16e7 - 1.16e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.50e7 + 1.50e7i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (9.74e6 + 9.74e6i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 - 1.46e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (-7.46e7 - 7.46e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 + 9.62e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-2.25e8 + 2.25e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 - 3.61e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (5.68e7 + 5.68e7i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 7.68e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (-7.95e8 - 7.95e8i)T + 7.60e17iT^{2} \) |
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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
−13.80292128533628818451571927102, −11.60016980582391110559991613690, −10.42772912215861124990480078815, −9.406851051795147812254476515205, −8.327400131895984082314744999021, −8.036290343507081320310880670874, −6.12910538529659638843653749586, −5.16792147057502097200284151763, −3.12876365075769268247417462327, −1.08874098707457150821977425113,
0.69977802943530740008139232377, 1.68405801010261974570962318888, 2.69006963741946580390366440733, 4.32962608396533053130885072959, 7.01592694900194855541383327922, 7.82077099012336979618649557551, 8.906070209095735091240652593548, 9.926131396579678910326915011761, 10.95584486513870546295367531311, 12.16873922151366646734369859852