L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.597 − 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.230 − 0.973i)5-s + (−0.993 + 0.116i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.396 + 0.918i)12-s + (−0.973 − 0.230i)13-s + (−0.973 − 0.230i)14-s + (−0.918 + 0.396i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.597 − 0.802i)3-s + (−0.5 − 0.866i)4-s + (0.230 − 0.973i)5-s + (−0.993 + 0.116i)6-s + (−0.286 − 0.957i)7-s − 8-s + (−0.286 + 0.957i)9-s + (−0.727 − 0.686i)10-s + (0.727 − 0.686i)11-s + (−0.396 + 0.918i)12-s + (−0.973 − 0.230i)13-s + (−0.973 − 0.230i)14-s + (−0.918 + 0.396i)15-s + (−0.5 + 0.866i)16-s + 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.5756858018 - 0.08320651232i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.5756858018 - 0.08320651232i\) |
\(L(1)\) |
\(\approx\) |
\(0.3256431322 - 0.8280681810i\) |
\(L(1)\) |
\(\approx\) |
\(0.3256431322 - 0.8280681810i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.597 - 0.802i)T \) |
| 5 | \( 1 + (0.230 - 0.973i)T \) |
| 7 | \( 1 + (-0.286 - 0.957i)T \) |
| 11 | \( 1 + (0.727 - 0.686i)T \) |
| 13 | \( 1 + (-0.973 - 0.230i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (0.230 - 0.973i)T \) |
| 31 | \( 1 + (-0.998 + 0.0581i)T \) |
| 41 | \( 1 + (0.342 + 0.939i)T \) |
| 43 | \( 1 + (-0.342 - 0.939i)T \) |
| 47 | \( 1 + (0.116 - 0.993i)T \) |
| 53 | \( 1 + (-0.802 + 0.597i)T \) |
| 59 | \( 1 + (-0.396 - 0.918i)T \) |
| 61 | \( 1 + (-0.549 + 0.835i)T \) |
| 67 | \( 1 + (-0.918 + 0.396i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.973 + 0.230i)T \) |
| 79 | \( 1 + (0.993 + 0.116i)T \) |
| 83 | \( 1 + (-0.893 + 0.448i)T \) |
| 89 | \( 1 + (-0.230 + 0.973i)T \) |
| 97 | \( 1 + (0.998 + 0.0581i)T \) |
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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
−18.79660746755511296384711681811, −18.25383130962022407542996902405, −17.624187318039258153433975579652, −16.85363847344222157089259112413, −16.44008524150469956523788093023, −15.65142957378069299077923163375, −14.87987406348056636700755019884, −14.53976728845058675735992902554, −14.251163506112090513379012012565, −12.72753103965683517247086877357, −12.30638944953485093538851869216, −11.82354772664774518330092199067, −10.84586101129321938194227114523, −9.98829043980239389035099741903, −9.45513644971936355462685473575, −8.84450677208293384607185502710, −7.721348760640606059877071731978, −7.01906590599157261750036470915, −6.26449783039115623204965422648, −5.867151873659465259322549042531, −5.03593446533540749221978046382, −4.37951787545563773846497260485, −3.422423589496278654095178273557, −2.94915657949328979625602681903, −1.82791624698617939167626238966,
0.167761324324720842066593155076, 0.966499867596301558953223079003, 1.49525886211072901841452147317, 2.50413447798768949742749263043, 3.44666685826702267536337403636, 4.28694698152111701827418248198, 5.04916312310794165463603566394, 5.61996943806375913480775723612, 6.356430430321414368130533083438, 7.229671402474957598306609895909, 7.9864959562556301359092352974, 8.97216479047591746125418039084, 9.66402567287484957409980858039, 10.3098174312006778201037027225, 11.18752265253187835069304160606, 11.73482259999142889168196568844, 12.361534225484450782648677008595, 12.986443257758712185299704131088, 13.56170389078399752818570188561, 13.97221144056198097256733301702, 14.84316546597683323502146989093, 15.91855203914549392276062517683, 16.77330009327567670271111438126, 17.171599793493514126252714285782, 17.711122652315494978186146523110