| L(s) = 1 | − 2-s + 3·3-s + 4-s + 5-s − 3·6-s + 4·7-s − 8-s + 6·9-s − 10-s + 5·11-s + 3·12-s − 3·13-s − 4·14-s + 3·15-s + 16-s − 4·17-s − 6·18-s + 19-s + 20-s + 12·21-s − 5·22-s − 23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s − 1.22·6-s + 1.51·7-s − 0.353·8-s + 2·9-s − 0.316·10-s + 1.50·11-s + 0.866·12-s − 0.832·13-s − 1.06·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s − 1.41·18-s + 0.229·19-s + 0.223·20-s + 2.61·21-s − 1.06·22-s − 0.208·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.141541792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.141541792\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−8.218366973672959523374228781211, −7.38272287944849707309785083867, −7.07057180715719522462714938151, −6.08332099416064060420385241288, −4.91487707863445356061052226296, −4.26262425560034400632469064199, −3.45938296129881677106992891237, −2.36364309361532861469806082298, −1.93945944010645515832197689160, −1.19241402941502560736094513471,
1.19241402941502560736094513471, 1.93945944010645515832197689160, 2.36364309361532861469806082298, 3.45938296129881677106992891237, 4.26262425560034400632469064199, 4.91487707863445356061052226296, 6.08332099416064060420385241288, 7.07057180715719522462714938151, 7.38272287944849707309785083867, 8.218366973672959523374228781211
