L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + 21-s − 23-s + (−0.309 − 0.951i)24-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)6-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (0.309 − 0.951i)17-s + (0.809 − 0.587i)18-s + 21-s − 23-s + (−0.309 − 0.951i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3429665136 + 0.04215966795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3429665136 + 0.04215966795i\) |
\(L(1)\) |
\(\approx\) |
\(0.4548600805 - 0.2420944032i\) |
\(L(1)\) |
\(\approx\) |
\(0.4548600805 - 0.2420944032i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 13 | \( 1 + (-0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.809 + 0.587i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 - 0.951i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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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
−21.87680307252536513399423759469, −20.86343393420598142587566690675, −19.71080943257989983986851891282, −19.08560711135058406810471938707, −18.15623208377893577686566774441, −17.32445852628715886251630743405, −16.73140338573600012563543865453, −16.21580708099084910906324452772, −15.48650431643691158748680704933, −14.61047527198421177118411483604, −13.83098501563859294849138918495, −12.84214694636383163963542677970, −12.02498679778995092295111560172, −10.83224141839716217578873938042, −10.15236996110709096596892693335, −9.53446169585495453490177437421, −8.69677310430059740064954050068, −7.48881663220829915117300817279, −6.71777203555932320399973222845, −6.06844597513559303626355842490, −5.23395302163544445132876545387, −4.158262511772334790760545061546, −3.66933362696989145048028768407, −1.66459471064338123885467229207, −0.24704737836385679077592904813,
0.85526618304564913751494492371, 2.12782144789846372403009793372, 2.88374406410455142618704620052, 3.99403219891259908495918356597, 5.25402688146794948079464901503, 5.773692298232011161499322739828, 7.068935968254565135860171008192, 7.78778916841854888412323373585, 8.82418021468702921860105418816, 9.78705421868169889618770593989, 10.38599893540885432645930863123, 11.34894368461731142042730790628, 12.082045691385163199127014084988, 12.61539973378782782820655544278, 13.30705915789712285109866401445, 14.14909128986421006927475523696, 15.511996133886287766074916217625, 16.32615380194318577350363266237, 17.12376749057658505741369682850, 17.86141851775092489574477261132, 18.635814574746116731558818700305, 18.987843003971627905963679338312, 20.01116218849914423279227453810, 20.60082141023963195426225471396, 21.81968324476723487957311540586