L(s) = 1 | + (−0.5 + 0.866i)3-s − 4-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + 16-s + (1.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)36-s + 1.73i·37-s + 0.999·39-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)48-s + (0.5 + 0.866i)52-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s − 4-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)12-s + (−0.5 − 0.866i)13-s + 16-s + (1.5 − 0.866i)19-s + (0.5 + 0.866i)25-s + 0.999·27-s + (0.499 + 0.866i)36-s + 1.73i·37-s + 0.999·39-s + (0.5 − 0.866i)43-s + (−0.5 + 0.866i)48-s + (0.5 + 0.866i)52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7282692094\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7282692094\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.73iT - T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.443509825291709453415471530647, −8.942857247351900771531994795596, −8.006726202537607326669702055287, −7.13088013422529407678909832241, −5.97975760862955154228519980021, −5.08270375959676737926421746361, −4.87708361920055056771955686231, −3.64692694192492682310858932750, −2.98584543240951196608292332664, −0.845312294892577637528653377864,
0.955850460410881885712347424039, 2.25026882348010017562896414617, 3.56822761450352521665703062324, 4.61213433406141015990881293163, 5.35251650774687348462217162307, 6.09768905813057962388267373373, 7.07993286125100485321485777199, 7.76089650629337650587065927261, 8.473562133858525512805613385050, 9.377586759013978429858816401509