L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.510 + 0.860i)3-s + (0.959 + 0.281i)4-s + (0.860 − 0.510i)5-s + (−0.382 − 0.923i)6-s + (−0.382 + 0.923i)7-s + (−0.909 − 0.415i)8-s + (−0.479 + 0.877i)9-s + (−0.923 + 0.382i)10-s + (0.654 + 0.755i)11-s + (0.247 + 0.968i)12-s + (0.177 − 0.984i)13-s + (0.510 − 0.860i)14-s + (0.877 + 0.479i)15-s + (0.841 + 0.540i)16-s + ⋯ |
L(s) = 1 | + (−0.989 − 0.142i)2-s + (0.510 + 0.860i)3-s + (0.959 + 0.281i)4-s + (0.860 − 0.510i)5-s + (−0.382 − 0.923i)6-s + (−0.382 + 0.923i)7-s + (−0.909 − 0.415i)8-s + (−0.479 + 0.877i)9-s + (−0.923 + 0.382i)10-s + (0.654 + 0.755i)11-s + (0.247 + 0.968i)12-s + (0.177 − 0.984i)13-s + (0.510 − 0.860i)14-s + (0.877 + 0.479i)15-s + (0.841 + 0.540i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08352384335 + 0.2908249694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08352384335 + 0.2908249694i\) |
\(L(1)\) |
\(\approx\) |
\(0.7290415363 + 0.2427244017i\) |
\(L(1)\) |
\(\approx\) |
\(0.7290415363 + 0.2427244017i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
| 353 | \( 1 \) |
good | 2 | \( 1 + (-0.989 - 0.142i)T \) |
| 3 | \( 1 + (0.510 + 0.860i)T \) |
| 5 | \( 1 + (0.860 - 0.510i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (0.177 - 0.984i)T \) |
| 19 | \( 1 + (-0.977 - 0.212i)T \) |
| 23 | \( 1 + (-0.0713 - 0.997i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.447 + 0.894i)T \) |
| 37 | \( 1 + (0.681 + 0.731i)T \) |
| 41 | \( 1 + (-0.997 - 0.0713i)T \) |
| 43 | \( 1 + (-0.0713 + 0.997i)T \) |
| 47 | \( 1 + (-0.877 + 0.479i)T \) |
| 53 | \( 1 + (-0.247 + 0.968i)T \) |
| 59 | \( 1 + (0.923 + 0.382i)T \) |
| 61 | \( 1 + (-0.841 - 0.540i)T \) |
| 67 | \( 1 + (-0.382 - 0.923i)T \) |
| 71 | \( 1 + (-0.778 - 0.627i)T \) |
| 73 | \( 1 + (0.959 + 0.281i)T \) |
| 79 | \( 1 + (-0.860 - 0.510i)T \) |
| 83 | \( 1 + (0.909 - 0.415i)T \) |
| 89 | \( 1 + (-0.681 + 0.731i)T \) |
| 97 | \( 1 + (0.909 - 0.415i)T \) |
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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
−17.33133192705171836389518018146, −16.850418204708001917581868809, −16.425336742544408155653800570159, −15.285311330383417796810398801441, −14.55977039310908433458755268409, −14.18678216699248012690435906500, −13.34394125246571209639827189605, −13.022653000961688533180984524327, −11.75083603381953625359271162269, −11.36936380447335532008760971321, −10.62172967978040272487124600719, −9.78541741290038581363034862650, −9.319216975285139055502979715927, −8.7179005855146992684986897126, −7.91118906429170109162818569673, −7.152033210561727402774440572376, −6.72491352444538336342260598665, −6.166296847782596218169840935913, −5.561121162081624118600880776875, −3.88592220406485374365999101870, −3.45343279664117591699487629848, −2.39416744752250130781877814436, −1.80051227765999579652725927363, −1.2123123287466172744931761220, −0.08810680533085703384715868767,
1.40542609723973790433677959763, 2.06397490706392338370315286798, 2.76212162291764673486208830825, 3.38723268811423381089302175375, 4.49946292491467154585740553118, 5.18954664736182817709036685971, 6.08761363166286834381018557247, 6.52350284691509674470692100972, 7.6616813627422898883518580528, 8.48214146659616601924392910688, 8.82841546228222339168466311375, 9.4613679704538221086092622141, 9.94233028154082103093193060424, 10.554592116854690499883782681888, 11.22674711229421163180127826337, 12.25734877875501700410040575399, 12.695209242132591906234386008925, 13.37204465372353381012311232721, 14.57391258325587587561633379030, 14.94184502051379626610476126227, 15.50756729871146146815691922301, 16.37628747394498430693185689381, 16.702366929010787567299324681974, 17.44034467610052661208249936222, 18.05903542812122805417904225892