L(s) = 1 | + (1.02 − 1.78i)2-s + (0.831 + 1.43i)3-s + (−1.11 − 1.93i)4-s + (0.5 − 0.866i)5-s + 3.41·6-s + (−1.49 − 2.18i)7-s − 0.473·8-s + (0.118 − 0.205i)9-s + (−1.02 − 1.78i)10-s + (0.5 + 0.866i)11-s + (1.85 − 3.21i)12-s + 1.53·13-s + (−5.42 + 0.423i)14-s + 1.66·15-s + (1.74 − 3.01i)16-s + (0.944 + 1.63i)17-s + ⋯ |
L(s) = 1 | + (0.727 − 1.25i)2-s + (0.479 + 0.831i)3-s + (−0.557 − 0.965i)4-s + (0.223 − 0.387i)5-s + 1.39·6-s + (−0.565 − 0.824i)7-s − 0.167·8-s + (0.0395 − 0.0685i)9-s + (−0.325 − 0.563i)10-s + (0.150 + 0.261i)11-s + (0.535 − 0.926i)12-s + 0.426·13-s + (−1.44 + 0.113i)14-s + 0.429·15-s + (0.435 − 0.754i)16-s + (0.229 + 0.396i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.140 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.75248 - 1.52101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.75248 - 1.52101i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.49 + 2.18i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.02 + 1.78i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.831 - 1.43i)T + (-1.5 + 2.59i)T^{2} \) |
| 13 | \( 1 - 1.53T + 13T^{2} \) |
| 17 | \( 1 + (-0.944 - 1.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.54 - 2.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.476 - 0.825i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 + (-1.88 - 3.27i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.20 - 2.08i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.20T + 41T^{2} \) |
| 43 | \( 1 + 3.36T + 43T^{2} \) |
| 47 | \( 1 + (3.84 - 6.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.824 + 1.42i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.66 - 6.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.53 - 9.58i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.95 - 3.39i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.15T + 71T^{2} \) |
| 73 | \( 1 + (4.71 + 8.16i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + (-6.09 + 10.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−11.03122180786097971817666685319, −10.21447022848493100806515952829, −9.765061635802627340785283089044, −8.767349634961390943640304268611, −7.41771980224508378182494337460, −6.00270956722771014870034892752, −4.62234375770366056589774282033, −3.90243848543765671639066127142, −3.16062328707634574668443088275, −1.48669928513565131059884301445,
2.13067113931931005371662344439, 3.50750500604948932401433766234, 5.02750578111443479020951033801, 6.04222824666231022665459067233, 6.70561110926631037321819567865, 7.55048286146622308596478783041, 8.407532816546623438236943642482, 9.372285644951569000929799427389, 10.67705093101439228654370509792, 11.84255899377033716653561619658