L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.841 + 0.540i)3-s + (−0.959 − 0.281i)4-s + (−0.654 + 0.755i)6-s + 1.51i·7-s + (0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s + 1.91·11-s + (−0.654 − 0.755i)12-s + (−1.49 − 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)18-s − 1.81i·19-s + (−0.817 + 1.27i)21-s + (−0.273 + 1.89i)22-s + ⋯ |
L(s) = 1 | + (−0.142 + 0.989i)2-s + (0.841 + 0.540i)3-s + (−0.959 − 0.281i)4-s + (−0.654 + 0.755i)6-s + 1.51i·7-s + (0.415 − 0.909i)8-s + (0.415 + 0.909i)9-s + 1.91·11-s + (−0.654 − 0.755i)12-s + (−1.49 − 0.215i)14-s + (0.841 + 0.540i)16-s + (−0.959 + 0.281i)18-s − 1.81i·19-s + (−0.817 + 1.27i)21-s + (−0.273 + 1.89i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.430650009\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.430650009\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 - 1.51iT - T^{2} \) |
| 11 | \( 1 - 1.91T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + 1.81iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.51iT - T^{2} \) |
| 31 | \( 1 + 1.97iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + 0.830T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.68T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 1.08iT - T^{2} \) |
| 97 | \( 1 + 1.91T + T^{2} \) |
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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
−9.195763714498531314714504022990, −9.026060903552162580174836380610, −8.267910459484283156478257617617, −7.31722973824093240994681937910, −6.50732065089162848271645693846, −5.73008358938635899917073877126, −4.80526574760662761237900985092, −4.06231288579216806573854395885, −3.03006468940317916439329839449, −1.77534703422381633789423465072,
1.19323653369796004732161807247, 1.76741432591720677145242629113, 3.33141579554695685553270990557, 3.85049662339810238304695147931, 4.40455664461373128219772250428, 6.05348827826622051144311936367, 6.88411230560012257326069125066, 7.74542586532691781422525631711, 8.303829557899426716957138633203, 9.218210375944351203552894297581