L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.826 + 0.563i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (0.955 − 0.294i)10-s + (0.623 − 0.781i)11-s + (0.0747 + 0.997i)12-s + (0.955 + 0.294i)13-s + (0.826 − 0.563i)14-s + (−0.988 + 0.149i)15-s + (−0.222 + 0.974i)16-s + (−0.733 − 0.680i)17-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.826 + 0.563i)3-s + (0.623 + 0.781i)4-s + (−0.733 + 0.680i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.222 − 0.974i)8-s + (0.365 + 0.930i)9-s + (0.955 − 0.294i)10-s + (0.623 − 0.781i)11-s + (0.0747 + 0.997i)12-s + (0.955 + 0.294i)13-s + (0.826 − 0.563i)14-s + (−0.988 + 0.149i)15-s + (−0.222 + 0.974i)16-s + (−0.733 − 0.680i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6019071077 + 0.2331826758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6019071077 + 0.2331826758i\) |
\(L(1)\) |
\(\approx\) |
\(0.7655876424 + 0.1524738964i\) |
\(L(1)\) |
\(\approx\) |
\(0.7655876424 + 0.1524738964i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (-0.900 - 0.433i)T \) |
| 3 | \( 1 + (0.826 + 0.563i)T \) |
| 5 | \( 1 + (-0.733 + 0.680i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.955 + 0.294i)T \) |
| 17 | \( 1 + (-0.733 - 0.680i)T \) |
| 19 | \( 1 + (0.365 - 0.930i)T \) |
| 23 | \( 1 + (-0.988 - 0.149i)T \) |
| 29 | \( 1 + (0.826 - 0.563i)T \) |
| 31 | \( 1 + (0.0747 + 0.997i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 47 | \( 1 + (0.623 + 0.781i)T \) |
| 53 | \( 1 + (0.955 - 0.294i)T \) |
| 59 | \( 1 + (-0.222 + 0.974i)T \) |
| 61 | \( 1 + (0.0747 - 0.997i)T \) |
| 67 | \( 1 + (0.365 - 0.930i)T \) |
| 71 | \( 1 + (-0.988 + 0.149i)T \) |
| 73 | \( 1 + (0.955 + 0.294i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.826 + 0.563i)T \) |
| 89 | \( 1 + (0.826 + 0.563i)T \) |
| 97 | \( 1 + (0.623 - 0.781i)T \) |
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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
−35.1690727568550672215671994553, −33.18677145222468076049730778134, −32.41809022754650976907837386444, −30.95138702578316208062848389136, −29.77228248258738255574269863030, −28.44636938277335740335975964302, −27.26587099189980965633680147453, −26.14911096093102759156593761533, −25.20658331988146734434412709802, −24.04875150425056773768840632621, −23.15053721534032075555175278308, −20.342772572598819944917162990282, −20.0717592747222592800325769888, −18.86532694366812845011164364354, −17.48635971980224012807316369442, −16.16871227611918924652338651453, −15.02485293585769421073382240828, −13.514858015302777328335759930019, −12.00092429281010479717529408880, −10.11152382814417587795531986584, −8.73514286029779617758797687686, −7.735936133521329107220922299369, −6.501738459213889005682048455025, −3.87772807411396579082970139048, −1.39909752181523223221470019462,
2.62523286655952035075974842607, 3.75816814825614387024191236852, 6.71548433181936992543005051613, 8.37472597309638274344771855324, 9.2169955810993646119576084162, 10.75035105593948037856763672479, 11.86804227566457767912495233933, 13.80718609333910882299293011705, 15.590017529971358466577824438872, 16.07158905360950267642983343013, 18.19913069436999997823070551365, 19.20630538006612677012928681673, 19.96677266510448469736257265474, 21.474911602967153480170094645, 22.34057302221512755772140568886, 24.56537773410241680145662937813, 25.75247125794290714376666475112, 26.60229788367684022953436265453, 27.53114542744559122737850235267, 28.639505248926069723416029826195, 30.28334439011530606059570457246, 31.03112053935256928899324799761, 32.26727118231562409588748676744, 33.873033918830098190314092897968, 35.02188403538532714266030818319