L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s + 2·13-s + 16-s + 2·17-s + 4·19-s − 4·22-s − 8·23-s + 2·26-s + 2·29-s + 8·31-s + 32-s + 2·34-s − 37-s + 4·38-s − 10·41-s − 12·43-s − 4·44-s − 8·46-s − 7·49-s + 2·52-s + 6·53-s + 2·58-s − 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s − 0.852·22-s − 1.66·23-s + 0.392·26-s + 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.342·34-s − 0.164·37-s + 0.648·38-s − 1.56·41-s − 1.82·43-s − 0.603·44-s − 1.17·46-s − 49-s + 0.277·52-s + 0.824·53-s + 0.262·58-s − 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.98578431063167, −15.57852708656249, −15.25841981426829, −14.35675956384819, −13.91188435630190, −13.47124923092883, −13.02325445338722, −12.22281594895743, −11.81836502902274, −11.38808197325735, −10.39407903689146, −10.20105554648108, −9.638143590256739, −8.491136045913202, −8.177355201931188, −7.582938397098639, −6.824360543249527, −6.168576601675747, −5.603746735629769, −4.966723373178621, −4.412377765252677, −3.414170394835028, −3.072875591530518, −2.123527912338604, −1.308868393941894, 0,
1.308868393941894, 2.123527912338604, 3.072875591530518, 3.414170394835028, 4.412377765252677, 4.966723373178621, 5.603746735629769, 6.168576601675747, 6.824360543249527, 7.582938397098639, 8.177355201931188, 8.491136045913202, 9.638143590256739, 10.20105554648108, 10.39407903689146, 11.38808197325735, 11.81836502902274, 12.22281594895743, 13.02325445338722, 13.47124923092883, 13.91188435630190, 14.35675956384819, 15.25841981426829, 15.57852708656249, 15.98578431063167