L(s) = 1 | + (0.208 − 0.977i)2-s + (−0.481 + 0.876i)3-s + (−0.912 − 0.408i)4-s + (0.393 − 0.919i)5-s + (0.756 + 0.653i)6-s + (0.997 − 0.0647i)7-s + (−0.590 + 0.807i)8-s + (−0.536 − 0.843i)9-s + (−0.816 − 0.577i)10-s + (0.0808 − 0.996i)11-s + (0.797 − 0.603i)12-s + (0.898 + 0.438i)13-s + (0.145 − 0.989i)14-s + (0.616 + 0.787i)15-s + (0.665 + 0.746i)16-s + (−0.816 − 0.577i)17-s + ⋯ |
L(s) = 1 | + (0.208 − 0.977i)2-s + (−0.481 + 0.876i)3-s + (−0.912 − 0.408i)4-s + (0.393 − 0.919i)5-s + (0.756 + 0.653i)6-s + (0.997 − 0.0647i)7-s + (−0.590 + 0.807i)8-s + (−0.536 − 0.843i)9-s + (−0.816 − 0.577i)10-s + (0.0808 − 0.996i)11-s + (0.797 − 0.603i)12-s + (0.898 + 0.438i)13-s + (0.145 − 0.989i)14-s + (0.616 + 0.787i)15-s + (0.665 + 0.746i)16-s + (−0.816 − 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7299914792 - 0.9911844727i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7299914792 - 0.9911844727i\) |
\(L(1)\) |
\(\approx\) |
\(0.9099633470 - 0.5411643695i\) |
\(L(1)\) |
\(\approx\) |
\(0.9099633470 - 0.5411643695i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 389 | \( 1 \) |
good | 2 | \( 1 + (0.208 - 0.977i)T \) |
| 3 | \( 1 + (-0.481 + 0.876i)T \) |
| 5 | \( 1 + (0.393 - 0.919i)T \) |
| 7 | \( 1 + (0.997 - 0.0647i)T \) |
| 11 | \( 1 + (0.0808 - 0.996i)T \) |
| 13 | \( 1 + (0.898 + 0.438i)T \) |
| 17 | \( 1 + (-0.816 - 0.577i)T \) |
| 19 | \( 1 + (-0.177 + 0.984i)T \) |
| 23 | \( 1 + (0.393 + 0.919i)T \) |
| 29 | \( 1 + (0.452 - 0.891i)T \) |
| 31 | \( 1 + (-0.302 - 0.953i)T \) |
| 37 | \( 1 + (-0.423 - 0.905i)T \) |
| 41 | \( 1 + (-0.177 - 0.984i)T \) |
| 43 | \( 1 + (0.712 + 0.701i)T \) |
| 47 | \( 1 + (-0.177 - 0.984i)T \) |
| 53 | \( 1 + (-0.957 + 0.287i)T \) |
| 59 | \( 1 + (-0.363 - 0.931i)T \) |
| 61 | \( 1 + (0.898 + 0.438i)T \) |
| 67 | \( 1 + (0.966 - 0.256i)T \) |
| 71 | \( 1 + (-0.912 - 0.408i)T \) |
| 73 | \( 1 + (0.665 - 0.746i)T \) |
| 79 | \( 1 + (-0.302 + 0.953i)T \) |
| 83 | \( 1 + (0.966 + 0.256i)T \) |
| 89 | \( 1 + (-0.986 - 0.161i)T \) |
| 97 | \( 1 + (-0.912 + 0.408i)T \) |
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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
−24.725240758751749269985929415303, −23.80021169102998827060055711956, −23.22435659882790474650031530303, −22.33999232948860206247813935867, −21.71111394351525674257904645493, −20.41643997956817258324505047564, −19.081584555703148576628441300617, −18.1022816250794174939175951348, −17.781683971356679331820435176056, −17.169475476378900715991843124813, −15.77262460767791414208520167487, −14.880277051224255241611162642357, −14.17562146060578645064093219562, −13.26578290583171498071063510283, −12.48290452752695745808969669150, −11.2175423925702086215295926801, −10.466359668790550979939674114560, −8.865556338178507337696341043930, −8.034355664097840038147162146550, −6.93557823123463849135729381047, −6.54359151578149300765157431170, −5.375939224787848818820465802030, −4.48492044696004417435634897333, −2.808669202601365282109405084388, −1.478969556852086126809905637438,
0.81054591368650076562961502367, 2.00588245617891126775674282867, 3.65863043341260019356479097608, 4.38279154520789647275025033391, 5.35557736609979108853428440345, 6.01229105567422513185947214883, 8.26233429424770719506498884454, 8.95246468806855467863253965403, 9.754518125824786970327950617315, 10.99944199806455193718835792775, 11.3491236845663864895426346702, 12.325499614702237636875492230187, 13.58859807261349567979936018364, 14.10830663410072721181585758230, 15.375115395650164896025132807627, 16.38340693134597186570750599146, 17.295483128263586611013711484447, 17.975513183074332364772812017058, 19.08054233152780028683648657250, 20.29977004966176117218611944071, 20.97652336108663505711388893260, 21.28045432198839000678451701103, 22.19127655943618010752675492778, 23.26100311584809810877156387035, 23.93149587359715465845720844970