L(s) = 1 | + (−0.5 + 0.133i)2-s + (0.133 − 0.5i)3-s + (−1.5 + 0.866i)4-s + 0.267i·6-s + (2.5 + 0.866i)7-s + (1.36 − 1.36i)8-s + (2.36 + 1.36i)9-s + (1.36 + 2.36i)11-s + (0.232 + 0.866i)12-s + (2 + 2i)13-s + (−1.36 − 0.0980i)14-s + (1.23 − 2.13i)16-s + (−3.73 − i)17-s + (−1.36 − 0.366i)18-s + (0.366 − 0.633i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.0947i)2-s + (0.0773 − 0.288i)3-s + (−0.750 + 0.433i)4-s + 0.109i·6-s + (0.944 + 0.327i)7-s + (0.482 − 0.482i)8-s + (0.788 + 0.455i)9-s + (0.411 + 0.713i)11-s + (0.0669 + 0.249i)12-s + (0.554 + 0.554i)13-s + (−0.365 − 0.0262i)14-s + (0.308 − 0.533i)16-s + (−0.905 − 0.242i)17-s + (−0.321 − 0.0862i)18-s + (0.0839 − 0.145i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.939704 + 0.268689i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.939704 + 0.268689i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.133i)T + (1.73 - i)T^{2} \) |
| 3 | \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.73 + i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.366 + 0.633i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (2.83 - 2.83i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (6.83 + 1.83i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (4.09 + 7.09i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.86 - 10.6i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + (-3.46 + 12.9i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-2.83 - 1.63i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.330 - 0.571i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.92 + 5.92i)T - 97iT^{2} \) |
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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
−12.88159004503712372725211459890, −11.89691330148397871461389820239, −10.80615907083101180519542025961, −9.555679502925952738676566369936, −8.724633377456072125069331342388, −7.75289709037691972528878450711, −6.83254941459970093343343053223, −4.97703755259523160912737068799, −4.09170186498259751481065436535, −1.81299987570860903987960866303,
1.30726378938097481718546695029, 3.83444985962074491034512330671, 4.83674537912692526457307710184, 6.17686119981450019069870709037, 7.74493522192712570841171594213, 8.712706971656801701001470968287, 9.548081958998004564952538252901, 10.66904077004593931355093531631, 11.25489075690703172988340066736, 12.80129006109182691949330571466