| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 4·7-s + 8-s + 9-s + 2·11-s − 12-s − 2·13-s − 4·14-s + 16-s + 17-s + 18-s + 8·19-s + 4·21-s + 2·22-s − 23-s − 24-s − 2·26-s − 27-s − 4·28-s − 4·29-s − 2·31-s + 32-s − 2·33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.872·21-s + 0.426·22-s − 0.208·23-s − 0.204·24-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 0.742·29-s − 0.359·31-s + 0.176·32-s − 0.348·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944943973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944943973\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−9.209755998454270114203911022719, −7.87105130903269111058989848290, −6.98710421223036875408666754803, −6.65493342530741298494903504210, −5.60597537364871891808742500625, −5.24911866627135529505220565858, −3.90552656055871966781048056829, −3.44276001183761516973585326928, −2.33700547962649024897939253258, −0.811083164846458355875009988219,
0.811083164846458355875009988219, 2.33700547962649024897939253258, 3.44276001183761516973585326928, 3.90552656055871966781048056829, 5.24911866627135529505220565858, 5.60597537364871891808742500625, 6.65493342530741298494903504210, 6.98710421223036875408666754803, 7.87105130903269111058989848290, 9.209755998454270114203911022719
