L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (−0.366 + 0.633i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−4.09 + 2.36i)11-s − 0.999i·12-s + (2.5 − 2.59i)13-s − 0.732·14-s + (−0.5 − 0.866i)16-s + (−1.96 − 1.13i)17-s + 0.999·18-s + (1.09 + 0.633i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (−0.138 + 0.239i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−1.23 + 0.713i)11-s − 0.288i·12-s + (0.693 − 0.720i)13-s − 0.195·14-s + (−0.125 − 0.216i)16-s + (−0.476 − 0.275i)17-s + 0.235·18-s + (0.251 + 0.145i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4102390943\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4102390943\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2.5 + 2.59i)T \) |
good | 7 | \( 1 + (0.366 - 0.633i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.09 - 2.36i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.96 + 1.13i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.633i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.36 - 3.09i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.23 - 2.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (5.23 + 9.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-9.86 + 5.69i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.63 + 3.83i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 + 0.464iT - 53T^{2} \) |
| 59 | \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.598 - 1.03i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.56 + 9.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.09 + 0.633i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 9.73T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + (2.19 - 1.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.042072503222052701644660260832, −7.970380386270347402625015405190, −7.54296081568450702977487115371, −6.51388147289340401254905336017, −5.68632840653515693347221698610, −5.21973842170854667271755696610, −4.24480508381831984261460304213, −3.33324269004847434731198677420, −2.12838362211561024869150728532, −0.14320750583377898379455311079,
1.27932508990451918092554064412, 2.46173035828381615471844561235, 3.45019214436369990884490863909, 4.47698057287811357499157176854, 5.21707547889817224781002011122, 6.20259266868804239162534777855, 6.66196012365266882884452137750, 7.944518095787652946297521639297, 8.467210547089387710548535874105, 9.512925034358081436774961104157