L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.766 + 0.642i)11-s + (−0.939 − 0.342i)12-s + (0.939 − 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.5 + 0.866i)3-s + (0.173 + 0.984i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)6-s + (0.5 − 0.866i)7-s + (−0.5 + 0.866i)8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.766 + 0.642i)11-s + (−0.939 − 0.342i)12-s + (0.939 − 0.342i)13-s + (0.939 − 0.342i)14-s + (−0.173 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4645931974 + 0.9672476183i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4645931974 + 0.9672476183i\) |
\(L(1)\) |
\(\approx\) |
\(0.8512802287 + 0.7978279313i\) |
\(L(1)\) |
\(\approx\) |
\(0.8512802287 + 0.7978279313i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 \) |
good | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.766 + 0.642i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.766 - 0.642i)T \) |
| 31 | \( 1 + (-0.173 + 0.984i)T \) |
| 37 | \( 1 + (0.766 - 0.642i)T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + (-0.766 - 0.642i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.766 + 0.642i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 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
−31.2728147591660400613844860535, −30.22165801128715767594442468319, −28.970633734358297757071310706972, −28.31026382744422772436643612105, −27.38627319390518292025712588171, −25.23419384242921476534086907775, −24.13151154712835290132577204454, −23.67110818559340328164351537681, −22.52770898354656565147990984161, −21.258703599371862132151200878003, −20.220264042489960534443398561, −18.86071163624187291273126988996, −18.347798253046601736912407755557, −16.396569648860598266436105548658, −15.34221867647587788615464562902, −13.71500817627322114797787006924, −12.84493400854463604031389821327, −11.605974609418195149760546961251, −11.2363352085244977543057643915, −8.99960023558562826633874907839, −7.55881202508332896333668015779, −5.789557094627061876237838912706, −4.92752509851659583223633327712, −2.974168433559201374524867461961, −1.211614497409214702136771987808,
3.38099826304543824827007064154, 4.297217919811487437211000281309, 5.63172151291345604504546700720, 7.126820282109225422550562090077, 8.24928838434111408590445828810, 10.403888925125762407928570395173, 11.25938914884399467915868489981, 12.59522629537839355444545836981, 14.211657475356471862671587128215, 15.14296423070222564079294160652, 16.004963539552691236884995338142, 17.105919153297162656394912231138, 18.26434617975847624110294435734, 20.36104432067221994421774965177, 21.033078692731196654115821304191, 22.44899103879053740904560084116, 23.20548747945078816100308375316, 23.76673491260658911920947668815, 25.60527167435883585368512247130, 26.55744576436288151186622789199, 27.24719266396296875516595884596, 28.65988425663432883944874064296, 30.23612784940538771009679492564, 30.87834669209433539402897828325, 32.12618511890937800037109731843