L(s) = 1 | + 1.73·2-s + 0.999·4-s + 1.41i·5-s − 2.44i·7-s − 1.73·8-s + 2.44i·10-s + (−1.73 + 2.82i)11-s − 4.89i·13-s − 4.24i·14-s − 5·16-s + 7.34i·19-s + 1.41i·20-s + (−2.99 + 4.89i)22-s − 2.82i·23-s + 2.99·25-s − 8.48i·26-s + ⋯ |
L(s) = 1 | + 1.22·2-s + 0.499·4-s + 0.632i·5-s − 0.925i·7-s − 0.612·8-s + 0.774i·10-s + (−0.522 + 0.852i)11-s − 1.35i·13-s − 1.13i·14-s − 1.25·16-s + 1.68i·19-s + 0.316i·20-s + (−0.639 + 1.04i)22-s − 0.589i·23-s + 0.599·25-s − 1.66i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58916 + 0.0524713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58916 + 0.0524713i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.73 - 2.82i)T \) |
good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7.34iT - 19T^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 4.89iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 12.2iT - 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−13.95844459681045818018906594389, −12.91221870578719934605318588220, −12.24895702682139049913130344600, −10.72374507669906852458487536860, −10.01958857385252337075061154060, −8.098278463691958770896283990155, −6.88648207920864990292020820073, −5.59184807281069130468428611784, −4.28026523619816053340617922409, −2.99297515248484655175350708632,
2.80058170977831227301855613318, 4.51514897553382018020429493092, 5.44408447847065810026241746411, 6.65327855109073829931799890249, 8.610143931275478110278774821258, 9.282488771616602212298812887487, 11.25033661634587718559180118348, 11.98080252994570488079927553944, 13.04359172855784564928119747738, 13.68642115610286660432311105395