L(s) = 1 | + (−0.978 + 0.207i)2-s + (−0.978 + 0.207i)3-s + (0.913 − 0.406i)4-s + (0.913 − 0.406i)6-s + (−0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)12-s + (0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.809 + 0.587i)18-s + (0.669 − 0.743i)21-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)24-s + ⋯ |
L(s) = 1 | + (−0.978 + 0.207i)2-s + (−0.978 + 0.207i)3-s + (0.913 − 0.406i)4-s + (0.913 − 0.406i)6-s + (−0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)12-s + (0.913 − 0.406i)13-s + (0.669 − 0.743i)14-s + (0.669 − 0.743i)16-s + (−0.104 + 0.994i)17-s + (−0.809 + 0.587i)18-s + (0.669 − 0.743i)21-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3728593138 - 0.3131005231i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3728593138 - 0.3131005231i\) |
\(L(1)\) |
\(\approx\) |
\(0.5013392476 + 0.03466180414i\) |
\(L(1)\) |
\(\approx\) |
\(0.5013392476 + 0.03466180414i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 3 | \( 1 + (-0.978 + 0.207i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.913 - 0.406i)T \) |
| 17 | \( 1 + (-0.104 + 0.994i)T \) |
| 23 | \( 1 + (0.913 - 0.406i)T \) |
| 29 | \( 1 + (0.913 - 0.406i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (-0.104 - 0.994i)T \) |
| 59 | \( 1 + (-0.978 + 0.207i)T \) |
| 61 | \( 1 + (-0.104 - 0.994i)T \) |
| 67 | \( 1 + (0.669 + 0.743i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.809 - 0.587i)T \) |
| 89 | \( 1 + (0.913 - 0.406i)T \) |
| 97 | \( 1 + (0.913 + 0.406i)T \) |
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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
−18.13985789135455842161986148722, −17.46290327107127386501312877904, −16.775401943647034067251246725030, −16.4174198605623212980801704933, −15.75175250296337005655470585391, −15.250488498261903579348408731807, −13.85751943128267215100138478370, −13.33984707123720407530040150871, −12.6410514949911312192713096843, −11.81698784331943764587028979726, −11.413789641888083306039458389076, −10.67780766479487537099592170653, −10.1069895688197154381702218699, −9.50115507133831699229362779865, −8.75060056440073988449757252122, −7.854114631398673684468343216944, −7.12692323630199817690072012616, −6.55019666982917302000265774596, −6.17406533780640076294278041823, −5.05809121323043986610610753601, −4.239174408670194429092062397328, −3.267385585916891276787295649449, −2.60486684051145582747620423515, −1.22963141663634067848309159400, −1.02371889117637824485142321460,
0.276187208842008842620636363894, 1.10962025962918450363013325197, 2.058464626257454858030799886284, 3.03549031112606018022423030415, 3.83082070191243736460892729662, 4.933818396316297673219898822544, 5.73417079208547853310119623511, 6.2831994063460986672249660765, 6.673350465685068030393624645595, 7.60528438623812101928462475795, 8.5211955839868038060963824185, 8.994529317763806377187707270599, 9.8406568676239985770040926432, 10.445526715246591540629409036997, 10.87999360890353427958083282929, 11.68752566090830593230674752108, 12.35698462632122694563617178046, 12.87146016827116803342944160223, 13.814872141445987460952083615577, 15.02104699283343981096065840467, 15.40671933063164274109887055237, 15.97873720810603066398398201995, 16.505066670931674162386245789150, 17.32066398571109494753674781307, 17.62400906208908096323816451302