L(s) = 1 | − 3.11·3-s + (0.660 − 2.13i)5-s + i·7-s + 6.70·9-s − 0.780i·11-s + 3.09·13-s + (−2.05 + 6.65i)15-s + 7.31i·17-s − 0.207i·19-s − 3.11i·21-s + 1.52i·23-s + (−4.12 − 2.82i)25-s − 11.5·27-s − 3.60i·29-s − 7.94·31-s + ⋯ |
L(s) = 1 | − 1.79·3-s + (0.295 − 0.955i)5-s + 0.377i·7-s + 2.23·9-s − 0.235i·11-s + 0.857·13-s + (−0.531 + 1.71i)15-s + 1.77i·17-s − 0.0476i·19-s − 0.679i·21-s + 0.318i·23-s + (−0.825 − 0.564i)25-s − 2.22·27-s − 0.669i·29-s − 1.42·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7432224543\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7432224543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.660 + 2.13i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 11 | \( 1 + 0.780iT - 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 - 7.31iT - 17T^{2} \) |
| 19 | \( 1 + 0.207iT - 19T^{2} \) |
| 23 | \( 1 - 1.52iT - 23T^{2} \) |
| 29 | \( 1 + 3.60iT - 29T^{2} \) |
| 31 | \( 1 + 7.94T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 - 6.61T + 43T^{2} \) |
| 47 | \( 1 + 2.61iT - 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 - 14.6iT - 59T^{2} \) |
| 61 | \( 1 - 6.37iT - 61T^{2} \) |
| 67 | \( 1 + 7.21T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 0.502iT - 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 10.2iT - 97T^{2} \) |
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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
−9.196652942696891394113588350304, −8.461146661322649454212815179646, −7.53676488375662272691654751310, −6.44476350991626595131468706780, −5.75910466658758174417001614656, −5.59355771098442891513155000172, −4.46269445594553343694518968693, −3.82219841115336296324073369829, −1.85580802064571461953259492688, −0.989770869300621474091884897020,
0.41375435168925578430830142060, 1.72183419669380381955548245884, 3.16379686667288419052371098843, 4.22754017993169835330132452004, 5.14693093864281462005095458839, 5.76543581297676517554734056904, 6.62066569760185620623967268258, 7.02463028649524246055835225660, 7.76582720716586372266296484197, 9.274603970975977353742747705081