L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s − 2·14-s − 15-s + 16-s + 6·17-s − 18-s + 2·19-s − 20-s + 2·21-s + 3·22-s + 3·23-s − 24-s + 25-s + 27-s + 2·28-s + 3·29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.639·22-s + 0.625·23-s − 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.797382445\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.797382445\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.228636712731349959298744028074, −7.66390626233387368721823594825, −7.21746323061510071028084070620, −6.17832098061906736409948413693, −5.22304312932503892335662858403, −4.63163589384940097057605522319, −3.40709762308697394376371776094, −2.87948152251303346299151316814, −1.77819381224379926635245867791, −0.818287666166974148659067464971,
0.818287666166974148659067464971, 1.77819381224379926635245867791, 2.87948152251303346299151316814, 3.40709762308697394376371776094, 4.63163589384940097057605522319, 5.22304312932503892335662858403, 6.17832098061906736409948413693, 7.21746323061510071028084070620, 7.66390626233387368721823594825, 8.228636712731349959298744028074