L(s) = 1 | + (0.296 + 0.955i)2-s + (−0.274 − 0.961i)3-s + (−0.824 + 0.565i)4-s + (−0.882 + 0.470i)5-s + (0.836 − 0.547i)6-s + (0.441 + 0.897i)7-s + (−0.784 − 0.619i)8-s + (−0.848 + 0.528i)9-s + (−0.711 − 0.703i)10-s + (−0.0779 + 0.996i)11-s + (0.770 + 0.637i)12-s + (0.0556 + 0.998i)13-s + (−0.726 + 0.687i)14-s + (0.695 + 0.718i)15-s + (0.359 − 0.933i)16-s + (0.958 − 0.285i)17-s + ⋯ |
L(s) = 1 | + (0.296 + 0.955i)2-s + (−0.274 − 0.961i)3-s + (−0.824 + 0.565i)4-s + (−0.882 + 0.470i)5-s + (0.836 − 0.547i)6-s + (0.441 + 0.897i)7-s + (−0.784 − 0.619i)8-s + (−0.848 + 0.528i)9-s + (−0.711 − 0.703i)10-s + (−0.0779 + 0.996i)11-s + (0.770 + 0.637i)12-s + (0.0556 + 0.998i)13-s + (−0.726 + 0.687i)14-s + (0.695 + 0.718i)15-s + (0.359 − 0.933i)16-s + (0.958 − 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1573063086 + 0.2113044776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1573063086 + 0.2113044776i\) |
\(L(1)\) |
\(\approx\) |
\(0.6303805569 + 0.3987854240i\) |
\(L(1)\) |
\(\approx\) |
\(0.6303805569 + 0.3987854240i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 283 | \( 1 \) |
good | 2 | \( 1 + (0.296 + 0.955i)T \) |
| 3 | \( 1 + (-0.274 - 0.961i)T \) |
| 5 | \( 1 + (-0.882 + 0.470i)T \) |
| 7 | \( 1 + (0.441 + 0.897i)T \) |
| 11 | \( 1 + (-0.0779 + 0.996i)T \) |
| 13 | \( 1 + (0.0556 + 0.998i)T \) |
| 17 | \( 1 + (0.958 - 0.285i)T \) |
| 19 | \( 1 + (0.892 + 0.451i)T \) |
| 23 | \( 1 + (-0.911 - 0.410i)T \) |
| 29 | \( 1 + (-0.944 - 0.328i)T \) |
| 31 | \( 1 + (-0.999 + 0.0445i)T \) |
| 37 | \( 1 + (-0.662 - 0.749i)T \) |
| 41 | \( 1 + (0.144 + 0.989i)T \) |
| 43 | \( 1 + (-0.920 + 0.390i)T \) |
| 47 | \( 1 + (-0.836 - 0.547i)T \) |
| 53 | \( 1 + (-0.359 - 0.933i)T \) |
| 59 | \( 1 + (-0.380 - 0.924i)T \) |
| 61 | \( 1 + (0.964 + 0.264i)T \) |
| 67 | \( 1 + (0.166 + 0.986i)T \) |
| 71 | \( 1 + (-0.824 - 0.565i)T \) |
| 73 | \( 1 + (0.999 + 0.0445i)T \) |
| 79 | \( 1 + (-0.920 - 0.390i)T \) |
| 83 | \( 1 + (0.951 - 0.306i)T \) |
| 89 | \( 1 + (-0.798 - 0.602i)T \) |
| 97 | \( 1 + (0.811 - 0.584i)T \) |
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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
−24.18096200955246119677384367790, −23.65998559912149337267046338757, −22.73878282327152816665844515197, −21.97680782413117123237149340838, −20.92477637197755904767776506525, −20.30114906793851147950080158064, −19.711969718220573570270553824777, −18.46755221816288757521690498761, −17.31787781468898159266307960703, −16.42169112988593680024236685675, −15.43127703347905971370776800771, −14.43943091267182355429884859386, −13.49885141119510973321750066617, −12.28523275430302284913618010130, −11.36988816195667923535823751949, −10.76173370575083329505991618867, −9.83778186860850668033823082842, −8.66901706644657859373456160352, −7.74070005629108016459045027097, −5.64863424732293335558355102410, −4.97282611210550697920507598365, −3.69622632397099320810337552880, −3.37023253472373627508337617407, −1.10369598892081211975029226055, −0.08849065308289740751533074047,
1.90901190360228190369498351236, 3.41661931965379569016145127677, 4.79739363616196998606104875116, 5.7978643716648729096233800352, 6.90295242474057304649599559683, 7.62168985866378890323620075974, 8.373365117709715734026265469034, 9.66573005732952657080190191787, 11.53902580110055771318035437121, 12.027101972424358744982501876201, 12.89530160275195482935116833500, 14.3608002262728392050925348711, 14.631244065617596862364362928328, 15.91837361977099158829435863942, 16.66496663523210952529997540832, 18.04459247879786048924982794520, 18.35161554471484934300283090449, 19.22474117780438428564423570013, 20.56890661507361570811210495373, 21.914268581370644536534373878519, 22.71598667878619561573534597951, 23.40036374279215164477733318725, 24.16430130944621191107831632547, 24.914209532188329427436995506214, 25.78240267503586849541446739809