| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 3·7-s − 8-s + 9-s − 10-s + 3·11-s − 12-s − 2·13-s − 3·14-s − 15-s + 16-s + 8·17-s − 18-s − 7·19-s + 20-s − 3·21-s − 3·22-s + 7·23-s + 24-s + 25-s + 2·26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 1.60·19-s + 0.223·20-s − 0.654·21-s − 0.639·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236264071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236264071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−10.08899779438703801843255855629, −9.268924861784239526218808930680, −8.465717536853394119135463223387, −7.52923302075403492535699943151, −6.80256742692428766584172992753, −5.72157748425877621726298098937, −5.01990188744444973228610877881, −3.74972054348016899323662400078, −2.12015702817127480647396497982, −1.09359249229865051341950577011,
1.09359249229865051341950577011, 2.12015702817127480647396497982, 3.74972054348016899323662400078, 5.01990188744444973228610877881, 5.72157748425877621726298098937, 6.80256742692428766584172992753, 7.52923302075403492535699943151, 8.465717536853394119135463223387, 9.268924861784239526218808930680, 10.08899779438703801843255855629
