L(s) = 1 | + (−0.5 − 0.866i)3-s + (−1.5 − 2.59i)5-s + (1 − 1.73i)9-s − 4·11-s + (2.5 − 4.33i)13-s + (−1.5 + 2.59i)15-s + (2.5 + 4.33i)17-s + (4 + 1.73i)19-s + (0.5 − 0.866i)23-s + (−2 + 3.46i)25-s − 5·27-s + (−1.5 + 2.59i)29-s + 4·31-s + (2 + 3.46i)33-s + 2·37-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (−0.670 − 1.16i)5-s + (0.333 − 0.577i)9-s − 1.20·11-s + (0.693 − 1.20i)13-s + (−0.387 + 0.670i)15-s + (0.606 + 1.05i)17-s + (0.917 + 0.397i)19-s + (0.104 − 0.180i)23-s + (−0.400 + 0.692i)25-s − 0.962·27-s + (−0.278 + 0.482i)29-s + 0.718·31-s + (0.348 + 0.603i)33-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.593257 - 0.654372i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.593257 - 0.654372i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-2.5 - 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.5 + 11.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.5 - 11.2i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.78901816022090662434005503767, −12.01255057718805074530326390348, −10.80376756342128163847906372351, −9.656916818908247081094457171789, −8.216548688067344586752233555432, −7.78111380016531325454854692236, −6.09566197658560200325766043446, −5.03494778594923827488258126283, −3.50875417154481424874690094537, −0.979914450721557854737958164682,
2.78405572274053258556836322948, 4.22027592841996732724125269361, 5.53051998006997938078980694943, 7.09112036315099228024375911223, 7.75348182049944111759256174068, 9.367134985478709320882469849148, 10.46177320669096020045510745729, 11.13253141962830958834015744702, 11.92785765795071117290117393839, 13.49623060698242654792520311593