L(s) = 1 | + (0.125 + 0.992i)2-s + (0.844 − 0.535i)3-s + (−0.968 + 0.248i)4-s + (0.637 + 0.770i)6-s + (0.587 + 0.809i)7-s + (−0.368 − 0.929i)8-s + (0.425 − 0.904i)9-s + (−0.684 + 0.728i)12-s + (−0.904 − 0.425i)13-s + (−0.728 + 0.684i)14-s + (0.876 − 0.481i)16-s + (−0.844 − 0.535i)17-s + (0.951 + 0.309i)18-s + (−0.637 − 0.770i)19-s + (0.929 + 0.368i)21-s + ⋯ |
L(s) = 1 | + (0.125 + 0.992i)2-s + (0.844 − 0.535i)3-s + (−0.968 + 0.248i)4-s + (0.637 + 0.770i)6-s + (0.587 + 0.809i)7-s + (−0.368 − 0.929i)8-s + (0.425 − 0.904i)9-s + (−0.684 + 0.728i)12-s + (−0.904 − 0.425i)13-s + (−0.728 + 0.684i)14-s + (0.876 − 0.481i)16-s + (−0.844 − 0.535i)17-s + (0.951 + 0.309i)18-s + (−0.637 − 0.770i)19-s + (0.929 + 0.368i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.384260785 - 0.5326992125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.384260785 - 0.5326992125i\) |
\(L(1)\) |
\(\approx\) |
\(1.201147394 + 0.1854919502i\) |
\(L(1)\) |
\(\approx\) |
\(1.201147394 + 0.1854919502i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.125 + 0.992i)T \) |
| 3 | \( 1 + (0.844 - 0.535i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 13 | \( 1 + (-0.904 - 0.425i)T \) |
| 17 | \( 1 + (-0.844 - 0.535i)T \) |
| 19 | \( 1 + (-0.637 - 0.770i)T \) |
| 23 | \( 1 + (0.481 - 0.876i)T \) |
| 29 | \( 1 + (0.0627 - 0.998i)T \) |
| 31 | \( 1 + (-0.929 + 0.368i)T \) |
| 37 | \( 1 + (-0.904 - 0.425i)T \) |
| 41 | \( 1 + (0.425 - 0.904i)T \) |
| 43 | \( 1 + (0.587 - 0.809i)T \) |
| 47 | \( 1 + (-0.770 - 0.637i)T \) |
| 53 | \( 1 + (0.770 + 0.637i)T \) |
| 59 | \( 1 + (0.187 + 0.982i)T \) |
| 61 | \( 1 + (0.187 - 0.982i)T \) |
| 67 | \( 1 + (0.844 + 0.535i)T \) |
| 71 | \( 1 + (0.535 + 0.844i)T \) |
| 73 | \( 1 + (-0.125 - 0.992i)T \) |
| 79 | \( 1 + (-0.929 - 0.368i)T \) |
| 83 | \( 1 + (-0.368 - 0.929i)T \) |
| 89 | \( 1 + (-0.728 + 0.684i)T \) |
| 97 | \( 1 + (0.248 + 0.968i)T \) |
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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
−21.055942368364233842482409870059, −20.04255076378628421971396973270, −19.71968916913833025190036587374, −19.01837531003623727146335367745, −18.0401157326495511572732039021, −17.206245287272143169162734713561, −16.5298598193704571468010857334, −15.289986255312836778920980865875, −14.50333061464772923435713913619, −14.20854605570780881753393112239, −13.17580204765193747879878702081, −12.68954486284480544735295691164, −11.411389366080156288753770158844, −10.883950168164928136726587125613, −10.08567899827102547442822134213, −9.44566364057996875402589101841, −8.58214595337573237660460961441, −7.88423767282208337926515541837, −6.90811620466473807996726214351, −5.386356448495077831661870882535, −4.57811135469876589469508418954, −3.99619506189335271968943335424, −3.169455891151866898727928960998, −2.08388750582661586593987194006, −1.4625896939614130307934673004,
0.46905578989315997144300730813, 2.10775750870470898222884820517, 2.76082065155923988647480454653, 4.02140124590365843299138696380, 4.86876756218702520876828185563, 5.70127107525418256308985676757, 6.805408257158264375186315925571, 7.278980779260268664168762906555, 8.23632417346446520348615633492, 8.8465159865637719356805333606, 9.35639463573505807642016477857, 10.54258920105411530438554992210, 11.83833374784887854381273045758, 12.58092822837284334365149497517, 13.20158921153146858424703081711, 14.09685736281253740026649131379, 14.710160441823442690939872204996, 15.31251066103467475888117141100, 15.871072735173123826258730319100, 17.19266194606669350754408802108, 17.6536968949802119298499366661, 18.4264059380809827809594004159, 19.05805773013252321441696148192, 19.86470356996560615156227425694, 20.8289948209814345320787741895