L(s) = 1 | + 2-s + 3-s + 4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + 13-s + 14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + 2-s + 3-s + 4-s + (−0.5 − 0.866i)5-s + 6-s + 7-s + 8-s + 9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 12-s + 13-s + 14-s + (−0.5 − 0.866i)15-s + 16-s + (−0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.826 - 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(4.184686056 - 1.288103532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.184686056 - 1.288103532i\) |
\(L(1)\) |
\(\approx\) |
\(2.695609455 - 0.4630356342i\) |
\(L(1)\) |
\(\approx\) |
\(2.695609455 - 0.4630356342i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−21.54221731313615741685713877135, −20.782389412086253732348877892494, −20.440756003344123353493563156317, −19.61008352003229970062414369216, −18.394460105703857593591152226295, −18.291274835845151141449483368172, −16.684480638902290656762116107089, −15.72092547309025868755199476499, −15.04691164021882647443193664840, −14.72085922268700401382293082637, −13.82018895730960836221488350485, −13.24768096999120678017412360818, −12.1788433922080890014285892947, −11.39749785438855450972543852122, −10.61179323647943747543868457979, −9.830330531430959621641473579623, −8.28775375477437684754472483909, −7.86821273875400906892282972699, −7.01089442140961012230202689111, −6.1728238343563064554029232926, −4.78562299774969997066493139360, −4.18077156851842923571217471702, −3.32887192709234945012331430515, −2.34191816540868094371773516885, −1.729923716857436934096100472193,
1.35301936701532352964784057353, 2.02932508906203277654331725201, 3.402654660951220935387765562663, 3.909338778412645662830053890278, 4.86283186988375182118647112698, 5.604714962367107833286271595584, 6.863571446730654331513264539710, 7.86365714789941406498656721274, 8.40827088133565990774639336195, 9.1017037694226813723145816062, 10.79784042242639694560207605224, 11.02757675677000957852851592250, 12.273783043294234486997877697752, 12.95366525962821322293858671555, 13.68109560743551918796657611946, 14.22205112368709620891916073791, 15.4232002550051166960598898082, 15.55021659792530484390282479039, 16.4953106750693678359105866177, 17.53847609396121218820925340775, 18.66961185544786359880758325639, 19.57419374836963906492553715616, 20.11804741309705184505434197163, 20.8661884805677068376320446828, 21.38809984033533964172440557051