L(s) = 1 | − 1.58·5-s + (2.64 − 0.0963i)7-s − 1.58·11-s + (2.40 + 4.16i)13-s + (2.69 + 4.67i)17-s + (3.54 − 6.14i)19-s + 0.300·23-s − 2.47·25-s + (−4.13 + 7.16i)29-s + (−1.35 + 2.34i)31-s + (−4.19 + 0.153i)35-s + (0.5 − 0.866i)37-s + (−2.93 − 5.08i)41-s + (0.833 − 1.44i)43-s + (−1.33 − 2.30i)47-s + ⋯ |
L(s) = 1 | − 0.710·5-s + (0.999 − 0.0364i)7-s − 0.478·11-s + (0.667 + 1.15i)13-s + (0.654 + 1.13i)17-s + (0.814 − 1.41i)19-s + 0.0626·23-s − 0.495·25-s + (−0.768 + 1.33i)29-s + (−0.243 + 0.421i)31-s + (−0.709 + 0.0258i)35-s + (0.0821 − 0.142i)37-s + (−0.458 − 0.794i)41-s + (0.127 − 0.220i)43-s + (−0.194 − 0.336i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.607590383\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.607590383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.64 + 0.0963i)T \) |
good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 + (-2.40 - 4.16i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.69 - 4.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.300T + 23T^{2} \) |
| 29 | \( 1 + (4.13 - 7.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.35 - 2.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.833 + 1.44i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.33 + 2.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.23 - 5.60i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.23 - 3.87i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.02 - 8.70i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.19 - 7.26i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.18 + 2.04i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.712 + 1.23i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.703531238310456693971455080753, −8.187316765281856381331793010556, −7.34399518887740712448712891694, −6.85026295796568211270608552199, −5.63633677427340783960001944506, −5.03115316498238278595075264563, −4.08927011698206686300301381354, −3.47019076826232470072981031272, −2.14377488277299951201098850220, −1.16308134385350254001313269882,
0.57286524031636062121677181299, 1.77874455956870688218544504916, 3.06523018624510409472988066047, 3.75033571057749233042028053187, 4.76970762639594834248807650437, 5.48144849853645460483724201638, 6.13209479122178023383706325298, 7.53032982242268679107008072497, 7.88204988504251607450098194690, 8.156466498680617640576016966929