L(s) = 1 | + 2·2-s + 2·4-s − 7-s + 4·13-s − 2·14-s − 4·16-s − 17-s − 4·23-s + 8·26-s − 2·28-s − 2·31-s − 8·32-s − 2·34-s − 6·37-s − 2·41-s + 3·43-s − 8·46-s − 7·47-s + 49-s + 8·52-s − 12·53-s − 5·59-s − 12·61-s − 4·62-s − 8·64-s − 5·67-s − 2·68-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.377·7-s + 1.10·13-s − 0.534·14-s − 16-s − 0.242·17-s − 0.834·23-s + 1.56·26-s − 0.377·28-s − 0.359·31-s − 1.41·32-s − 0.342·34-s − 0.986·37-s − 0.312·41-s + 0.457·43-s − 1.17·46-s − 1.02·47-s + 1/7·49-s + 1.10·52-s − 1.64·53-s − 0.650·59-s − 1.53·61-s − 0.508·62-s − 64-s − 0.610·67-s − 0.242·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.291194376\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.291194376\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.11947233283585, −12.77274799670746, −12.26509510032364, −11.87873421316206, −11.37960979519277, −10.85118842519862, −10.54967518710259, −9.765874695950472, −9.377046273686069, −8.769151485493253, −8.396297830066942, −7.741105991469039, −7.120781420213006, −6.638606985714132, −6.060124785697017, −5.906671232651060, −5.267453057547478, −4.612787970900726, −4.261553059806012, −3.689672551130773, −3.125706019502895, −2.881171473311659, −1.835185329659290, −1.543856857313468, −0.3111694438952727,
0.3111694438952727, 1.543856857313468, 1.835185329659290, 2.881171473311659, 3.125706019502895, 3.689672551130773, 4.261553059806012, 4.612787970900726, 5.267453057547478, 5.906671232651060, 6.060124785697017, 6.638606985714132, 7.120781420213006, 7.741105991469039, 8.396297830066942, 8.769151485493253, 9.377046273686069, 9.765874695950472, 10.54967518710259, 10.85118842519862, 11.37960979519277, 11.87873421316206, 12.26509510032364, 12.77274799670746, 13.11947233283585