L(s) = 1 | − 9.86·5-s + 2.64i·7-s + 13.1i·11-s − 5.86·13-s + 0.570·17-s + 15.6i·19-s + 16.4i·23-s + 72.3·25-s + 29.7·29-s + 54.8i·31-s − 26.1i·35-s − 42.0·37-s − 0.773·41-s − 41.7i·43-s + 58.4i·47-s + ⋯ |
L(s) = 1 | − 1.97·5-s + 0.377i·7-s + 1.19i·11-s − 0.451·13-s + 0.0335·17-s + 0.824i·19-s + 0.716i·23-s + 2.89·25-s + 1.02·29-s + 1.76i·31-s − 0.745i·35-s − 1.13·37-s − 0.0188·41-s − 0.971i·43-s + 1.24i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2290112121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2290112121\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
good | 5 | \( 1 + 9.86T + 25T^{2} \) |
| 11 | \( 1 - 13.1iT - 121T^{2} \) |
| 13 | \( 1 + 5.86T + 169T^{2} \) |
| 17 | \( 1 - 0.570T + 289T^{2} \) |
| 19 | \( 1 - 15.6iT - 361T^{2} \) |
| 23 | \( 1 - 16.4iT - 529T^{2} \) |
| 29 | \( 1 - 29.7T + 841T^{2} \) |
| 31 | \( 1 - 54.8iT - 961T^{2} \) |
| 37 | \( 1 + 42.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 0.773T + 1.68e3T^{2} \) |
| 43 | \( 1 + 41.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.65T + 2.80e3T^{2} \) |
| 59 | \( 1 - 42.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 95.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 69.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 92.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 9.97T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 151. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.79T + 7.92e3T^{2} \) |
| 97 | \( 1 - 103.T + 9.40e3T^{2} \) |
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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
−9.310494679815914206329071743149, −8.540149343127648743228639581536, −7.82061664323433935156673047472, −7.27754130871114393707060132226, −6.57789677644600111727790138427, −5.16632234423602671055992528357, −4.57981165354464053442934879732, −3.70789804139042745582462458317, −2.91463483658251957255966941653, −1.46751387908249705463144284599,
0.085709610164939527483237803853, 0.792504659276148726884460527571, 2.73026124910802363250251780844, 3.51623264768618148489714139785, 4.30175850719558041254623781510, 4.98836440730888110615512403855, 6.27370267159365344976224437091, 7.08081646015066529698218112188, 7.75222955173482174891921799962, 8.398309307562416587100945597024