L(s) = 1 | + (1.20 + 0.741i)2-s + 2.57·3-s + (0.901 + 1.78i)4-s + (−0.460 − 2.18i)5-s + (3.10 + 1.91i)6-s + (−2.31 + 1.28i)7-s + (−0.236 + 2.81i)8-s + 3.65·9-s + (1.06 − 2.97i)10-s − 3.59·11-s + (2.32 + 4.60i)12-s − 1.33i·13-s + (−3.73 − 0.172i)14-s + (−1.18 − 5.64i)15-s + (−2.37 + 3.21i)16-s + 2.88·17-s + ⋯ |
L(s) = 1 | + (0.851 + 0.523i)2-s + 1.48·3-s + (0.450 + 0.892i)4-s + (−0.205 − 0.978i)5-s + (1.26 + 0.780i)6-s + (−0.874 + 0.484i)7-s + (−0.0837 + 0.996i)8-s + 1.21·9-s + (0.337 − 0.941i)10-s − 1.08·11-s + (0.671 + 1.32i)12-s − 0.370i·13-s + (−0.998 − 0.0460i)14-s + (−0.306 − 1.45i)15-s + (−0.593 + 0.804i)16-s + 0.699·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.55680 + 0.831617i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.55680 + 0.831617i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.741i)T \) |
| 5 | \( 1 + (0.460 + 2.18i)T \) |
| 7 | \( 1 + (2.31 - 1.28i)T \) |
good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 + 1.33iT - 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 19 | \( 1 + 5.38iT - 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 + 1.88iT - 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 - 4.64T + 37T^{2} \) |
| 41 | \( 1 - 0.606iT - 41T^{2} \) |
| 43 | \( 1 - 12.1iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 6.31T + 53T^{2} \) |
| 59 | \( 1 + 1.39iT - 59T^{2} \) |
| 61 | \( 1 - 3.80T + 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 8.19iT - 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 7.04iT - 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−12.62385607629284822170757272962, −11.23981740902392106364848988938, −9.631679191153869255903216203072, −8.886531097280970964920846088864, −8.039944969092104423190050422268, −7.30226128938268972029672397074, −5.77045518758691838527981115517, −4.72489127524140514270686003354, −3.37387468173456938521165578721, −2.56879701027096687596796982909,
2.25415947737710379310224673053, 3.27983186012872760767537475508, 3.83367620664058515209733761921, 5.63537736786977074017404662397, 7.00208943651984364001941094365, 7.65966853828555893485196406473, 9.100638009027547141237436345267, 10.16692222940059498119432741856, 10.56310176456687959178281849876, 11.93689673917659063450502774061