L(s) = 1 | + (0.951 − 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (−0.587 + 0.809i)23-s + (0.587 − 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.951 + 0.309i)33-s + (0.587 + 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)3-s − i·7-s + (0.809 − 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.309 + 0.951i)21-s + (−0.587 + 0.809i)23-s + (0.587 − 0.809i)27-s + (−0.309 − 0.951i)29-s + (−0.309 + 0.951i)31-s + (0.951 + 0.309i)33-s + (0.587 + 0.809i)37-s + (−0.809 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.344010530 - 0.04223723248i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.344010530 - 0.04223723248i\) |
\(L(1)\) |
\(\approx\) |
\(1.332519553 - 0.03963542506i\) |
\(L(1)\) |
\(\approx\) |
\(1.332519553 - 0.03963542506i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.587 - 0.809i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.587 + 0.809i)T \) |
| 29 | \( 1 + (-0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.951 + 0.309i)T \) |
| 59 | \( 1 + (-0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.951 + 0.309i)T \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.587 - 0.809i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (-0.951 - 0.309i)T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (0.951 - 0.309i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.05041653627839790949167955348, −29.12741853549087757811313216044, −27.56497246349771649123732700575, −26.712066529658055408026095775059, −26.09277140790248426627748321358, −24.70791683207060199967017404443, −24.02421035999960779441560374367, −22.46651147883687013780326049324, −21.52481445073532608300723572087, −20.30457380606400699848308741534, −19.67721791941591693799553070508, −18.55657371369434894647881062753, −16.9431474574376889082272391715, −16.16486266566674886706415387900, −14.59503147731974968971500435768, −14.06304879518695349567600644838, −12.85664163643860368610602909188, −11.24464802671502252890933414243, −10.02505014427654943409177181004, −8.9737294379066012579957777209, −7.755010774003055668450372642186, −6.54768160312709283897155587515, −4.48073852212963786376546929943, −3.574781326000960575390329380239, −1.827673324264072119471936429497,
1.92875516039207501279766894184, 3.12731275590550216267690687717, 4.78122236343968103402597778414, 6.486745264046017039204599190286, 7.72496915799651067854093639661, 8.9599053431575930350993413611, 9.75801724716987796765338642782, 11.63430814477296189633952558976, 12.66072538095144967204631890456, 13.77418575163566803571514704779, 15.04754098801139674736431781707, 15.57927278528974942963367523898, 17.47084500326566153824703174831, 18.33403930378635978831524393316, 19.618994905522756081943140306782, 20.17833602930944671870716442538, 21.60956514299968542264095553538, 22.43299828841005229064069633119, 24.04425117619190345683261722894, 24.926905513336786189766507102135, 25.5507053556061478000642747737, 26.79957802453597471593944286939, 27.77885428002611447113362234848, 28.96838168445643838330924738523, 30.21787230497937245022156172844