| L(s) = 1 | − 2-s + 3-s + 4-s − 3·5-s − 6-s − 7-s − 8-s + 9-s + 3·10-s + 2·11-s + 12-s + 3·13-s + 14-s − 3·15-s + 16-s − 17-s − 18-s − 3·20-s − 21-s − 2·22-s + 5·23-s − 24-s + 4·25-s − 3·26-s + 27-s − 28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.603·11-s + 0.288·12-s + 0.832·13-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.670·20-s − 0.218·21-s − 0.426·22-s + 1.04·23-s − 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.429054579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.429054579\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.83690403975773, −12.16818869773471, −11.88460413542206, −11.39461157946643, −10.98151935159486, −10.56996891026064, −9.985076867350957, −9.457730920300921, −8.903879662977684, −8.749461335340064, −8.193216652704863, −7.699242729468600, −7.356623562073190, −6.752076401586176, −6.475193715473014, −5.780927396466818, −5.117831156681817, −4.369431765007538, −3.919216278120189, −3.631031813715128, −2.848547066753119, −2.604079734180889, −1.551773638120334, −1.104471217163266, −0.3966033339801131,
0.3966033339801131, 1.104471217163266, 1.551773638120334, 2.604079734180889, 2.848547066753119, 3.631031813715128, 3.919216278120189, 4.369431765007538, 5.117831156681817, 5.780927396466818, 6.475193715473014, 6.752076401586176, 7.356623562073190, 7.699242729468600, 8.193216652704863, 8.749461335340064, 8.903879662977684, 9.457730920300921, 9.985076867350957, 10.56996891026064, 10.98151935159486, 11.39461157946643, 11.88460413542206, 12.16818869773471, 12.83690403975773
