L(s) = 1 | + (−0.133 − 0.5i)2-s + (0.5 + 0.133i)3-s + (1.5 − 0.866i)4-s + (1.86 − 1.23i)5-s − 0.267i·6-s + (−1.36 − 1.36i)8-s + (−2.36 − 1.36i)9-s + (−0.866 − 0.767i)10-s + (1.36 + 2.36i)11-s + (0.866 − 0.232i)12-s + (−2 + 2i)13-s + (1.09 − 0.366i)15-s + (1.23 − 2.13i)16-s + (−1 + 3.73i)17-s + (−0.366 + 1.36i)18-s + (0.366 − 0.633i)19-s + ⋯ |
L(s) = 1 | + (−0.0947 − 0.353i)2-s + (0.288 + 0.0773i)3-s + (0.750 − 0.433i)4-s + (0.834 − 0.550i)5-s − 0.109i·6-s + (−0.482 − 0.482i)8-s + (−0.788 − 0.455i)9-s + (−0.273 − 0.242i)10-s + (0.411 + 0.713i)11-s + (0.249 − 0.0669i)12-s + (−0.554 + 0.554i)13-s + (0.283 − 0.0945i)15-s + (0.308 − 0.533i)16-s + (−0.242 + 0.905i)17-s + (−0.0862 + 0.321i)18-s + (0.0839 − 0.145i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41768 - 0.732527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41768 - 0.732527i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.86 + 1.23i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.133 + 0.5i)T + (-1.73 + i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.133i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-1.36 - 2.36i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 - 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 - 3.73i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.366 + 0.633i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.133 - 0.0358i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3iT - 29T^{2} \) |
| 31 | \( 1 + (-6.46 + 3.73i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.26 - 4.73i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 6.46iT - 41T^{2} \) |
| 43 | \( 1 + (-2.83 - 2.83i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.83 - 2.36i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2} \) |
| 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 | \( 1 + (10.6 + 2.86i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 + (-12.9 - 3.46i)T + (63.2 + 36.5i)T^{2} \) |
| 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
−11.95309456707407308937991085448, −11.01351922728638060348829532745, −9.804761413160760257189877989698, −9.412721123783757448262158240975, −8.200595171564925584220776174621, −6.67722601813082860408384906143, −5.98975151675537099346997134530, −4.59663359487590538473910174108, −2.83583777330632344433760511590, −1.61174520549788935873907986488,
2.34402929176256115520639832499, 3.19843128648142824645905099243, 5.35017392651604571999591095560, 6.26303189106745977186327868055, 7.25128396601842605418829092890, 8.225338497739255309866223726014, 9.201547376408162869584491347098, 10.42499640619369446405023620188, 11.26805802128253656866594859045, 12.09764153305846473317456099502