L(s) = 1 | + (−1.20 + 0.742i)2-s + (−0.505 + 1.65i)3-s + (0.897 − 1.78i)4-s + (−2.00 − 2.00i)5-s + (−0.621 − 2.36i)6-s + 7-s + (0.246 + 2.81i)8-s + (−2.48 − 1.67i)9-s + (3.90 + 0.925i)10-s + (2.67 − 2.67i)11-s + (2.50 + 2.39i)12-s + (4.14 + 4.14i)13-s + (−1.20 + 0.742i)14-s + (4.34 − 2.31i)15-s + (−2.38 − 3.20i)16-s + 3.45i·17-s + ⋯ |
L(s) = 1 | + (−0.851 + 0.525i)2-s + (−0.291 + 0.956i)3-s + (0.448 − 0.893i)4-s + (−0.897 − 0.897i)5-s + (−0.253 − 0.967i)6-s + 0.377·7-s + (0.0872 + 0.996i)8-s + (−0.829 − 0.558i)9-s + (1.23 + 0.292i)10-s + (0.806 − 0.806i)11-s + (0.723 + 0.690i)12-s + (1.14 + 1.14i)13-s + (−0.321 + 0.198i)14-s + (1.12 − 0.596i)15-s + (−0.597 − 0.802i)16-s + 0.838i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.714286 + 0.275647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.714286 + 0.275647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.20 - 0.742i)T \) |
| 3 | \( 1 + (0.505 - 1.65i)T \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (2.00 + 2.00i)T + 5iT^{2} \) |
| 11 | \( 1 + (-2.67 + 2.67i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.14 - 4.14i)T + 13iT^{2} \) |
| 17 | \( 1 - 3.45iT - 17T^{2} \) |
| 19 | \( 1 + (-5.27 + 5.27i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.11iT - 23T^{2} \) |
| 29 | \( 1 + (-0.808 + 0.808i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.56iT - 31T^{2} \) |
| 37 | \( 1 + (1.06 - 1.06i)T - 37iT^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + (5.32 + 5.32i)T + 43iT^{2} \) |
| 47 | \( 1 - 6.19T + 47T^{2} \) |
| 53 | \( 1 + (0.414 + 0.414i)T + 53iT^{2} \) |
| 59 | \( 1 + (-7.26 + 7.26i)T - 59iT^{2} \) |
| 61 | \( 1 + (1.06 + 1.06i)T + 61iT^{2} \) |
| 67 | \( 1 + (-4.81 + 4.81i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.83iT - 71T^{2} \) |
| 73 | \( 1 + 0.150iT - 73T^{2} \) |
| 79 | \( 1 + 4.73iT - 79T^{2} \) |
| 83 | \( 1 + (1.94 + 1.94i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−11.37682089991000465196469308157, −10.86801532256524241821701234016, −9.471699837137615216838321699740, −8.801554052957668998473595297319, −8.335358137761221872106870849170, −6.89771954494297334675038898541, −5.80044332694621397746987407952, −4.73144405650535402512186065934, −3.68430538801712609465925777347, −1.04530500404167165063380663139,
1.10615253697352849533203057301, 2.74036959420213754675244640894, 3.89504813387289884579659407635, 5.89390904719991641699128928830, 7.08532538379328928979838256917, 7.64365991110780546940229910300, 8.372779879428243631223992707650, 9.696562025375595960736067516219, 10.79615678592054512276004827239, 11.43479719962392838153991461209