L(s) = 1 | + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (2.21 + 0.314i)5-s + (0.707 − 0.707i)6-s + (−0.376 − 0.376i)7-s − i·8-s + 1.00i·9-s + (−0.314 + 2.21i)10-s − 1.28i·11-s + (0.707 + 0.707i)12-s − 5.12i·13-s + (0.376 − 0.376i)14-s + (−1.34 − 1.78i)15-s + 16-s − 2.31·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 − 0.408i)3-s − 0.5·4-s + (0.990 + 0.140i)5-s + (0.288 − 0.288i)6-s + (−0.142 − 0.142i)7-s − 0.353i·8-s + 0.333i·9-s + (−0.0994 + 0.700i)10-s − 0.387i·11-s + (0.204 + 0.204i)12-s − 1.42i·13-s + (0.100 − 0.100i)14-s + (−0.346 − 0.461i)15-s + 0.250·16-s − 0.560·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.412007760\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.412007760\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 + (-2.21 - 0.314i)T \) |
| 37 | \( 1 + (-5.17 + 3.20i)T \) |
good | 7 | \( 1 + (0.376 + 0.376i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.28iT - 11T^{2} \) |
| 13 | \( 1 + 5.12iT - 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 + (-3.87 - 3.87i)T + 19iT^{2} \) |
| 23 | \( 1 + 6.29iT - 23T^{2} \) |
| 29 | \( 1 + (1.15 - 1.15i)T - 29iT^{2} \) |
| 31 | \( 1 + (7.11 + 7.11i)T + 31iT^{2} \) |
| 41 | \( 1 - 2.12iT - 41T^{2} \) |
| 43 | \( 1 + 2.56iT - 43T^{2} \) |
| 47 | \( 1 + (-2.64 - 2.64i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.572 + 0.572i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.04 + 1.04i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.44 - 3.44i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.32 + 5.32i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 + (-9.81 - 9.81i)T + 73iT^{2} \) |
| 79 | \( 1 + (11.3 + 11.3i)T + 79iT^{2} \) |
| 83 | \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \) |
| 89 | \( 1 + (-8.09 + 8.09i)T - 89iT^{2} \) |
| 97 | \( 1 - 8.07T + 97T^{2} \) |
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
−9.781979019841591237923316082555, −8.876658541247433015612227767559, −7.958407584473889401060928206443, −7.23763939800619433933959725747, −6.21744737687382063568461895008, −5.76686079510810868902436008724, −4.98097419326474437677844830049, −3.57490651125929325303373877852, −2.29095735820632380069943120291, −0.70520680952504279863003701665,
1.39936317621450217362367317968, 2.44715428359783785674309428813, 3.70335426766002465463441042134, 4.77297632082858154855205430734, 5.40791215546452749846607380361, 6.46570943758619874490080162908, 7.27133922500401998091720402651, 8.753581454689467657543726328364, 9.511576912138875678699210324134, 9.612904421249985600222309795896