| L(s) = 1 | + 2-s + 2·3-s + 4-s + 5-s + 2·6-s + 4·7-s + 8-s + 9-s + 10-s − 4·11-s + 2·12-s + 4·13-s + 4·14-s + 2·15-s + 16-s − 4·17-s + 18-s + 4·19-s + 20-s + 8·21-s − 4·22-s + 6·23-s + 2·24-s + 25-s + 4·26-s − 4·27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.577·12-s + 1.10·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 1.74·21-s − 0.852·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s + 0.784·26-s − 0.769·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.838073009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.838073009\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.478698812851939208558315303758, −7.76219914973933315725457055417, −7.16869644960114806695480938160, −6.08742464413894673568726937334, −5.23211364759196044704322927943, −4.78989476874132783058571155116, −3.74403785987274033633942007586, −2.91665557873146578482513089779, −2.20019267522323475884301546271, −1.35836847603348318164409695699,
1.35836847603348318164409695699, 2.20019267522323475884301546271, 2.91665557873146578482513089779, 3.74403785987274033633942007586, 4.78989476874132783058571155116, 5.23211364759196044704322927943, 6.08742464413894673568726937334, 7.16869644960114806695480938160, 7.76219914973933315725457055417, 8.478698812851939208558315303758
