L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.597 + 0.802i)3-s + (−0.939 − 0.342i)4-s + (−0.957 − 0.286i)5-s + (−0.686 − 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.835 − 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.597 + 0.802i)3-s + (−0.939 − 0.342i)4-s + (−0.957 − 0.286i)5-s + (−0.686 − 0.727i)6-s + (−0.686 − 0.727i)7-s + (0.5 − 0.866i)8-s + (−0.286 − 0.957i)9-s + (0.448 − 0.893i)10-s + (−0.448 − 0.893i)11-s + (0.835 − 0.549i)12-s + (−0.973 + 0.230i)13-s + (0.835 − 0.549i)14-s + (0.802 − 0.597i)15-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5013460366 - 0.03606459078i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5013460366 - 0.03606459078i\) |
\(L(1)\) |
\(\approx\) |
\(0.4822976105 + 0.1899471071i\) |
\(L(1)\) |
\(\approx\) |
\(0.4822976105 + 0.1899471071i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (-0.597 + 0.802i)T \) |
| 5 | \( 1 + (-0.957 - 0.286i)T \) |
| 7 | \( 1 + (-0.686 - 0.727i)T \) |
| 11 | \( 1 + (-0.448 - 0.893i)T \) |
| 13 | \( 1 + (-0.973 + 0.230i)T \) |
| 17 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.448 + 0.893i)T \) |
| 31 | \( 1 + (0.918 + 0.396i)T \) |
| 41 | \( 1 + (-0.642 + 0.766i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (0.230 - 0.973i)T \) |
| 53 | \( 1 + (0.549 + 0.835i)T \) |
| 59 | \( 1 + (-0.597 - 0.802i)T \) |
| 61 | \( 1 + (-0.448 + 0.893i)T \) |
| 67 | \( 1 + (0.998 + 0.0581i)T \) |
| 71 | \( 1 + (-0.173 - 0.984i)T \) |
| 73 | \( 1 + (0.0581 - 0.998i)T \) |
| 79 | \( 1 + (-0.893 + 0.448i)T \) |
| 83 | \( 1 + (0.286 - 0.957i)T \) |
| 89 | \( 1 + (0.918 - 0.396i)T \) |
| 97 | \( 1 + (0.802 + 0.597i)T \) |
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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
−18.692361353987499554718215637844, −18.08782124337583044919086086258, −17.19291834326443515887967783116, −16.79653407236314463682065788070, −15.76405138999128818941202440610, −15.02551455541702828627620091046, −14.37076289264288414646089623436, −13.28662263643848216290518904333, −12.592621261466582487973150578158, −12.39230677555432953161780244860, −11.75751155988656491973819385562, −11.09936012628101229537937692510, −10.1585484854448989776389889749, −9.87286175546218174267650859292, −8.65857800826529940856594058628, −7.99969889132340739825889693325, −7.468627759745480575021712610911, −6.61894773678987162960964925702, −5.68147017969761832498629615173, −4.92122134009815841043858183581, −4.16654578045881644572488059312, −3.1416944499855302853858449948, −2.52127407534174746851488189344, −1.80024431298646201219718880417, −0.62543996261649773792219639847,
0.35384369796591136077171871390, 0.93430576217302978171747520465, 3.16399565358774354589194529210, 3.42068029857175772175574382040, 4.58537329985774614610486464925, 4.917338285093308929745452492379, 5.604311891651568061621029208043, 6.71863651906381901971059977059, 7.12332475272946204877532044646, 7.82456437950878407018707929943, 8.89048500517128920364487002049, 9.25929178160768905280169797984, 10.10657856332125923767848574544, 10.72307359084012083405520804260, 11.5619059462660125367761091784, 12.20029092632948597773678157692, 13.199532444971373583609775986, 13.72942903975964980287079889340, 14.632535311157482809780216460796, 15.38054744734900330335093637368, 15.81333827324751238308324466518, 16.44031260279845711986286505399, 16.814254932998302163558844340343, 17.42291212923414027171838772498, 18.40476419879136557826931834652