L(s) = 1 | + (−0.906 − 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (0.0241 − 0.999i)6-s + (0.568 − 0.822i)7-s + (−0.262 − 0.964i)8-s + (−0.681 + 0.732i)9-s + (−0.607 + 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (−0.861 + 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (−0.443 − 0.896i)17-s + (0.926 − 0.377i)18-s + ⋯ |
L(s) = 1 | + (−0.906 − 0.421i)2-s + (0.399 + 0.916i)3-s + (0.644 + 0.764i)4-s + (0.215 − 0.976i)5-s + (0.0241 − 0.999i)6-s + (0.568 − 0.822i)7-s + (−0.262 − 0.964i)8-s + (−0.681 + 0.732i)9-s + (−0.607 + 0.794i)10-s + (−0.443 + 0.896i)12-s + (0.568 − 0.822i)13-s + (−0.861 + 0.506i)14-s + (0.981 − 0.192i)15-s + (−0.168 + 0.985i)16-s + (−0.443 − 0.896i)17-s + (0.926 − 0.377i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3008094415 - 0.6944884890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3008094415 - 0.6944884890i\) |
\(L(1)\) |
\(\approx\) |
\(0.7290283892 - 0.2134535033i\) |
\(L(1)\) |
\(\approx\) |
\(0.7290283892 - 0.2134535033i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.906 - 0.421i)T \) |
| 3 | \( 1 + (0.399 + 0.916i)T \) |
| 5 | \( 1 + (0.215 - 0.976i)T \) |
| 7 | \( 1 + (0.568 - 0.822i)T \) |
| 13 | \( 1 + (0.568 - 0.822i)T \) |
| 17 | \( 1 + (-0.443 - 0.896i)T \) |
| 19 | \( 1 + (-0.989 - 0.144i)T \) |
| 23 | \( 1 + (-0.998 + 0.0483i)T \) |
| 29 | \( 1 + (0.120 + 0.992i)T \) |
| 31 | \( 1 + (0.836 - 0.548i)T \) |
| 37 | \( 1 + (-0.970 - 0.239i)T \) |
| 41 | \( 1 + (0.715 + 0.698i)T \) |
| 43 | \( 1 + (0.215 - 0.976i)T \) |
| 47 | \( 1 + (-0.681 + 0.732i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.443 - 0.896i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 + (0.215 + 0.976i)T \) |
| 71 | \( 1 + (0.779 + 0.626i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.861 - 0.506i)T \) |
| 83 | \( 1 + (0.644 - 0.764i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.943 + 0.331i)T \) |
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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
−21.07414223827602276663296863968, −19.81789926913842527727067445717, −19.173999081110433997653617269088, −18.77704884357918821066162412205, −17.957371539586761584275660567142, −17.645263704243286300505805145144, −16.71967934091952976917475211675, −15.401645960178000474312991497594, −15.15679828545355436218190051445, −14.16088787637230187063491974846, −13.80406668671687188565862172035, −12.44283347536795080852508920639, −11.67406881078655785443790951429, −11.00107151307603359919183636527, −10.15099871046011472269406652954, −9.11084037671101681043233432247, −8.45049353660813631930301548115, −7.89441798538954914878348973008, −6.89198290474100202357356870839, −6.22931240405347572719601468960, −5.82830709068435374614329407315, −4.20513001875261532982027203839, −2.823899531446915011459057309259, −2.05629233536373059110286850189, −1.545285413052898642548212127885,
0.34449320020325853539292125349, 1.51632590522295689575173667281, 2.47724358373499555779024118597, 3.597111865969676608548401273162, 4.33172403135543008502407640257, 5.13791075078436974544110310701, 6.32119605889315387113952686085, 7.58328846468625678069110880510, 8.27758563897807117498385508357, 8.778731849535016890700146914386, 9.64738850950233280428408574412, 10.33008831271534851621632206715, 10.96563897469509068709733428837, 11.74210954347897585959658103100, 12.81371387660699443588773343463, 13.52584973706260862078463957995, 14.37170933193263207213464190169, 15.54745771787330083186445990025, 16.01642083101032564673263384903, 16.73525806238906814178125394236, 17.46747594287599246904721951104, 17.93068767868959746710116249985, 19.18385710407717173813109712003, 19.99627606685463263302721332129, 20.4021543131727761765489034715