L(s) = 1 | + (0.987 − 0.156i)5-s + (−0.278 + 0.142i)13-s + (−0.0966 − 0.610i)17-s + (0.951 − 0.309i)25-s + (1.44 − 1.04i)29-s + (0.809 + 1.58i)37-s + (0.297 + 0.0966i)41-s − i·49-s + (0.253 − 1.59i)53-s + (0.363 + 1.11i)61-s + (−0.253 + 0.183i)65-s + (−0.896 + 1.76i)73-s + (−0.190 − 0.587i)85-s + (−0.280 − 0.863i)89-s + (−0.142 + 0.896i)97-s + ⋯ |
L(s) = 1 | + (0.987 − 0.156i)5-s + (−0.278 + 0.142i)13-s + (−0.0966 − 0.610i)17-s + (0.951 − 0.309i)25-s + (1.44 − 1.04i)29-s + (0.809 + 1.58i)37-s + (0.297 + 0.0966i)41-s − i·49-s + (0.253 − 1.59i)53-s + (0.363 + 1.11i)61-s + (−0.253 + 0.183i)65-s + (−0.896 + 1.76i)73-s + (−0.190 − 0.587i)85-s + (−0.280 − 0.863i)89-s + (−0.142 + 0.896i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.558424095\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.558424095\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.987 + 0.156i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.0966 + 0.610i)T + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-1.44 + 1.04i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.297 - 0.0966i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.253 + 1.59i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (0.280 + 0.863i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.634256702496578359964412948964, −8.144542947074620731019452773347, −7.03470931841089139660419814590, −6.51151025366769416783117780821, −5.69636956741191479602287649149, −4.95209201410929965107565414093, −4.23675505622649651157135596616, −2.94711416351828900849009082652, −2.28560742611870771667674937585, −1.08674627387388856452880654864,
1.25494633908653020092728810319, 2.31964874293552945137074925577, 3.08959879392439105421695318858, 4.22369428574666282775610964581, 5.04945559546314270248417641515, 5.86200271173667826508656373689, 6.41899314208079912373906780063, 7.24810176523421917165293298448, 8.010338798298492953876954052764, 8.966194523570235642863645928198