L(s) = 1 | − 2-s + 3-s + 4-s + 3.90·5-s − 6-s + 2.61·7-s − 8-s + 9-s − 3.90·10-s + 4.06·11-s + 12-s − 3.58·13-s − 2.61·14-s + 3.90·15-s + 16-s − 7.39·17-s − 18-s + 3.90·20-s + 2.61·21-s − 4.06·22-s + 3.11·23-s − 24-s + 10.2·25-s + 3.58·26-s + 27-s + 2.61·28-s − 2.95·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.74·5-s − 0.408·6-s + 0.989·7-s − 0.353·8-s + 0.333·9-s − 1.23·10-s + 1.22·11-s + 0.288·12-s − 0.994·13-s − 0.699·14-s + 1.00·15-s + 0.250·16-s − 1.79·17-s − 0.235·18-s + 0.872·20-s + 0.571·21-s − 0.865·22-s + 0.649·23-s − 0.204·24-s + 2.04·25-s + 0.703·26-s + 0.192·27-s + 0.494·28-s − 0.547·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.552375059\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.552375059\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 - 2.61T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 3.58T + 13T^{2} \) |
| 17 | \( 1 + 7.39T + 17T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 + 2.95T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 - 5.60T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 - 2.95T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 4.45T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 1.56T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 - 2.93T + 97T^{2} \) |
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
−9.197581825627117981742131372306, −8.578622424604711204770930591673, −7.60271753487786438628556105978, −6.71590720471283450401866526407, −6.20924753824730726651968851430, −5.06576303399819193056567093476, −4.32601961781086582138875981394, −2.70842284175534463632136476898, −2.06792956497804443284985125099, −1.28614092520257004600179100776,
1.28614092520257004600179100776, 2.06792956497804443284985125099, 2.70842284175534463632136476898, 4.32601961781086582138875981394, 5.06576303399819193056567093476, 6.20924753824730726651968851430, 6.71590720471283450401866526407, 7.60271753487786438628556105978, 8.578622424604711204770930591673, 9.197581825627117981742131372306