L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.142 − 0.989i)3-s + (−0.654 + 0.755i)4-s + (0.841 − 0.540i)6-s − 1.08i·7-s + (−0.959 − 0.281i)8-s + (−0.959 + 0.281i)9-s + 1.30·11-s + (0.841 + 0.540i)12-s + (0.983 − 0.449i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)18-s − 0.563i·19-s + (−1.07 + 0.153i)21-s + (0.544 + 1.19i)22-s + ⋯ |
L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.142 − 0.989i)3-s + (−0.654 + 0.755i)4-s + (0.841 − 0.540i)6-s − 1.08i·7-s + (−0.959 − 0.281i)8-s + (−0.959 + 0.281i)9-s + 1.30·11-s + (0.841 + 0.540i)12-s + (0.983 − 0.449i)14-s + (−0.142 − 0.989i)16-s + (−0.654 − 0.755i)18-s − 0.563i·19-s + (−1.07 + 0.153i)21-s + (0.544 + 1.19i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.190431511\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.190431511\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.142 + 0.989i)T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + 1.08iT - T^{2} \) |
| 11 | \( 1 - 1.30T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 0.563iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.08iT - T^{2} \) |
| 31 | \( 1 + 1.81iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.91T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.284T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.97iT - T^{2} \) |
| 97 | \( 1 + 1.30T + T^{2} \) |
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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
−9.099945123526947744198126803519, −8.132576618358243388597939307112, −7.54108521378009775109095094792, −6.92353123945210441463524168047, −6.25130226029610824114877687755, −5.58508673427531047986895518467, −4.29228105616542668826509004898, −3.80668576115319999058876370672, −2.39591821652881302389623929964, −0.821356546807012871468183666687,
1.58219327958329133255244849674, 2.79893431239952689024205794906, 3.64255122552273147148191816058, 4.35009334602545203064514680570, 5.35484805609351314044853233179, 5.82707997005026235943988095459, 6.76464643771582641455260546907, 8.361252873877966315066126284997, 9.013196219139209236541804679348, 9.373198838361257665282238493173