L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 6·11-s − 12-s − 14-s + 15-s + 16-s − 6·17-s + 18-s + 8·19-s − 20-s + 21-s + 6·22-s − 24-s + 25-s − 27-s − 28-s − 6·29-s + 30-s − 6·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.27·22-s − 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.182·30-s − 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.97387375650940, −13.42921069791260, −12.87767507218053, −12.26279211477349, −12.02826315750208, −11.55056613675241, −11.10844755303044, −10.77346288984646, −9.980968992962969, −9.470399184599770, −8.938390742425730, −8.690246932335736, −7.550398292813925, −7.237288347448858, −6.930209020958495, −6.185175123295208, −5.946216505387127, −5.114841665356467, −4.807593901220502, −3.911587974125185, −3.755844836536223, −3.219215862310078, −2.232567920609283, −1.616260207443861, −0.9294441300077686, 0,
0.9294441300077686, 1.616260207443861, 2.232567920609283, 3.219215862310078, 3.755844836536223, 3.911587974125185, 4.807593901220502, 5.114841665356467, 5.946216505387127, 6.185175123295208, 6.930209020958495, 7.237288347448858, 7.550398292813925, 8.690246932335736, 8.938390742425730, 9.470399184599770, 9.980968992962969, 10.77346288984646, 11.10844755303044, 11.55056613675241, 12.02826315750208, 12.26279211477349, 12.87767507218053, 13.42921069791260, 13.97387375650940