L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (0.951 − 0.309i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + i·12-s + (0.587 − 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.587 + 0.809i)17-s + (−0.951 + 0.309i)18-s + (0.309 − 0.951i)19-s − 21-s + ⋯ |
L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.951 − 0.309i)3-s + (−0.309 − 0.951i)4-s + (0.809 − 0.587i)6-s + (0.951 − 0.309i)7-s + (0.951 + 0.309i)8-s + (0.809 + 0.587i)9-s + i·12-s + (0.587 − 0.809i)13-s + (−0.309 + 0.951i)14-s + (−0.809 + 0.587i)16-s + (0.587 + 0.809i)17-s + (−0.951 + 0.309i)18-s + (0.309 − 0.951i)19-s − 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5503282578 + 0.06006559246i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5503282578 + 0.06006559246i\) |
\(L(1)\) |
\(\approx\) |
\(0.6508393320 + 0.09579938490i\) |
\(L(1)\) |
\(\approx\) |
\(0.6508393320 + 0.09579938490i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.587 + 0.809i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 13 | \( 1 + (0.587 - 0.809i)T \) |
| 17 | \( 1 + (0.587 + 0.809i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (0.309 + 0.951i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.951 + 0.309i)T \) |
| 53 | \( 1 + (-0.587 + 0.809i)T \) |
| 59 | \( 1 + (-0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.809 - 0.587i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.809 + 0.587i)T \) |
| 73 | \( 1 + (-0.951 + 0.309i)T \) |
| 79 | \( 1 + (-0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.587 - 0.809i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−33.468039070610996698856180472021, −31.71800091889851299853320338655, −30.60489507704438487106455275473, −29.46320726209551709904727506834, −28.54288850840372639791911574897, −27.552574378625779646171785447442, −26.88885155734373490287118306015, −25.321038915294105907814249326214, −23.75454133536486658095223032828, −22.55064019834404266787481752143, −21.327354863213187222358069403, −20.73939072726247016378280882456, −18.886465466451958601455326766998, −18.02966278089450555364328300559, −16.98333566315694287664696984133, −15.83829864921251847921921212137, −13.91980790937408921773074116794, −12.137346284523891890557672614056, −11.48606060504573254322808598522, −10.296984907341494325431348299019, −8.97505960390645510222355316244, −7.37511120664568264868424324156, −5.374053218279447064050464018109, −3.87749190417858927367172114376, −1.576080194815700384002094861592,
1.26959791632776242148227469179, 4.721092282288538746306197032125, 5.894312604463391549488023422141, 7.262392577107812315261258607162, 8.41608252142690754550326423991, 10.316405811168342519409553276597, 11.203122543034017308760966191823, 12.980721655873408015297192238706, 14.46207866133616822788094775616, 15.78745689388763468654311484011, 16.99813772665704048460146264250, 17.81013543486729266904562299457, 18.72745929299690548947901030783, 20.29739022898702522895838471467, 22.00943660701152490758765551264, 23.31550644798732076554593923303, 24.012290634046437396228216666536, 25.06262273896952541362880866865, 26.47157905840968837762246234263, 27.704698222162852117745505886018, 28.22496393647885817073566762062, 29.6696737130935714119221168650, 30.78293586622161980847311911549, 32.657635888987720611772534603338, 33.30613409542745829584308529346