L(s) = 1 | + (−0.957 − 0.288i)2-s + (0.0365 − 0.999i)3-s + (0.833 + 0.551i)4-s + (−0.391 + 0.920i)5-s + (−0.322 + 0.946i)6-s + (0.872 − 0.489i)7-s + (−0.639 − 0.768i)8-s + (−0.997 − 0.0729i)9-s + (0.639 − 0.768i)10-s + (−0.905 − 0.424i)11-s + (0.581 − 0.813i)12-s + (0.957 + 0.288i)13-s + (−0.976 + 0.217i)14-s + (0.905 + 0.424i)15-s + (0.391 + 0.920i)16-s + (0.791 − 0.611i)17-s + ⋯ |
L(s) = 1 | + (−0.957 − 0.288i)2-s + (0.0365 − 0.999i)3-s + (0.833 + 0.551i)4-s + (−0.391 + 0.920i)5-s + (−0.322 + 0.946i)6-s + (0.872 − 0.489i)7-s + (−0.639 − 0.768i)8-s + (−0.997 − 0.0729i)9-s + (0.639 − 0.768i)10-s + (−0.905 − 0.424i)11-s + (0.581 − 0.813i)12-s + (0.957 + 0.288i)13-s + (−0.976 + 0.217i)14-s + (0.905 + 0.424i)15-s + (0.391 + 0.920i)16-s + (0.791 − 0.611i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.203 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5722258709 - 0.4656640561i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5722258709 - 0.4656640561i\) |
\(L(1)\) |
\(\approx\) |
\(0.6719150131 - 0.2882059633i\) |
\(L(1)\) |
\(\approx\) |
\(0.6719150131 - 0.2882059633i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.957 - 0.288i)T \) |
| 3 | \( 1 + (0.0365 - 0.999i)T \) |
| 5 | \( 1 + (-0.391 + 0.920i)T \) |
| 7 | \( 1 + (0.872 - 0.489i)T \) |
| 11 | \( 1 + (-0.905 - 0.424i)T \) |
| 13 | \( 1 + (0.957 + 0.288i)T \) |
| 17 | \( 1 + (0.791 - 0.611i)T \) |
| 19 | \( 1 + (0.976 + 0.217i)T \) |
| 23 | \( 1 + (0.905 - 0.424i)T \) |
| 29 | \( 1 + (-0.322 - 0.946i)T \) |
| 31 | \( 1 + (-0.0365 - 0.999i)T \) |
| 37 | \( 1 + (-0.976 - 0.217i)T \) |
| 41 | \( 1 + (-0.872 + 0.489i)T \) |
| 43 | \( 1 + (0.833 - 0.551i)T \) |
| 47 | \( 1 + (0.989 - 0.145i)T \) |
| 53 | \( 1 + (-0.252 - 0.967i)T \) |
| 59 | \( 1 + (-0.520 + 0.853i)T \) |
| 61 | \( 1 + (0.791 + 0.611i)T \) |
| 67 | \( 1 + (-0.0365 + 0.999i)T \) |
| 71 | \( 1 + (0.322 + 0.946i)T \) |
| 73 | \( 1 + (-0.181 - 0.983i)T \) |
| 79 | \( 1 + (-0.989 - 0.145i)T \) |
| 83 | \( 1 + (-0.694 + 0.719i)T \) |
| 89 | \( 1 + (0.109 - 0.994i)T \) |
| 97 | \( 1 + (0.694 + 0.719i)T \) |
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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
−27.724242963582585245454806788512, −26.9618257654920119572020285623, −25.841850952876023109751765715639, −25.11526715810165447999835698369, −23.931135574357974618071868479061, −23.19142180738564918025648476520, −21.440372387461593438846210661268, −20.72232538778436005848126662511, −20.17782660621375104801027508414, −18.828748952975177273858254973714, −17.7358740941271461090938377732, −16.88351488099942868004949463685, −15.71962051714882993611206901203, −15.47964968077207091420256949849, −14.17548916717930915874011742206, −12.39965456474128176861450063061, −11.26971243084890735720721574762, −10.45363454629760881977113249825, −9.18460045823212589860490705306, −8.48953802199588338057695438046, −7.607663626267625604401417685419, −5.56014257982260803397262233322, −5.02050916177916736819581068, −3.23992309625744862957357312136, −1.42061575671651626262732829237,
0.94071450651963791591520181573, 2.3850494581519363291506924771, 3.4946172399278731816910464307, 5.77767408987720132746855224747, 7.1440972537617131171951301170, 7.69588190233467609980406475056, 8.61648574299304417610566040778, 10.26824240496606421612359721470, 11.25757725185674997008379715969, 11.76944089027065031900512530491, 13.30417593313074629277986360271, 14.27900310766520548300451710140, 15.56652540358994097668962546642, 16.77287361059330556159100445931, 17.86930214512050168830372198288, 18.61557765097256620700623846086, 19.00061361638908551991970662917, 20.43317963214816714490957361964, 20.99636374682601596982360587749, 22.64192467718762489365121693597, 23.610203509849076804877279862491, 24.46970665955675868280805492686, 25.57774650600494197693420332896, 26.40505074405287855806265754558, 27.12887042057825332192144606605