L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 2·14-s + 15-s + 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 2·21-s + 22-s − 3·23-s − 24-s + 25-s − 27-s + 2·28-s − 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 + 0.301·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s − 0.185·29-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
\(L(1)\) |
\(\approx\) |
\(2.797994403\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.797994403\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.86535315719615194384882014925, −7.52722004753888617611227254432, −6.77027353878417493667883014148, −5.77555978098150802290344658639, −5.40803780879719792858262290796, −4.52926317045913549612335328046, −3.91335537884438089797798211050, −3.04600236012356684594928960197, −1.87526134432411664682973508379, −0.882755338603577860882373327797,
0.882755338603577860882373327797, 1.87526134432411664682973508379, 3.04600236012356684594928960197, 3.91335537884438089797798211050, 4.52926317045913549612335328046, 5.40803780879719792858262290796, 5.77555978098150802290344658639, 6.77027353878417493667883014148, 7.52722004753888617611227254432, 7.86535315719615194384882014925