| L(s) = 1 | − 2-s + 3-s + 4-s − 4·5-s − 6-s + 2·7-s − 8-s + 9-s + 4·10-s − 11-s + 12-s − 2·14-s − 4·15-s + 16-s − 17-s − 18-s − 4·20-s + 2·21-s + 22-s − 6·23-s − 24-s + 11·25-s + 27-s + 2·28-s − 2·29-s + 4·30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.288·12-s − 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.894·20-s + 0.436·21-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.730·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7077851056\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7077851056\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.97303454978115, −12.45991993520033, −12.13530105610746, −11.55255089136676, −11.31424846712103, −10.72296500225335, −10.42446286065043, −9.747924437462895, −9.160531734162231, −8.599845448241798, −8.437937255430236, −7.788961190394617, −7.492405465246648, −7.308974434225179, −6.496551739148740, −5.915008229217520, −5.159391887891432, −4.588764430803640, −4.110258642894278, −3.625831253189206, −3.116513148585275, −2.376345231331114, −1.836341925552410, −1.079627611934610, −0.2851302531404390,
0.2851302531404390, 1.079627611934610, 1.836341925552410, 2.376345231331114, 3.116513148585275, 3.625831253189206, 4.110258642894278, 4.588764430803640, 5.159391887891432, 5.915008229217520, 6.496551739148740, 7.308974434225179, 7.492405465246648, 7.788961190394617, 8.437937255430236, 8.599845448241798, 9.160531734162231, 9.747924437462895, 10.42446286065043, 10.72296500225335, 11.31424846712103, 11.55255089136676, 12.13530105610746, 12.45991993520033, 12.97303454978115
