L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.142 + 0.989i)10-s + (−0.415 − 0.909i)11-s + (0.415 + 0.909i)12-s + (−0.142 − 0.989i)13-s + (−0.841 − 0.540i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + ⋯ |
L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.841 + 0.540i)5-s + (−0.654 + 0.755i)6-s + (0.142 − 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.142 + 0.989i)10-s + (−0.415 − 0.909i)11-s + (0.415 + 0.909i)12-s + (−0.142 − 0.989i)13-s + (−0.841 − 0.540i)14-s + (0.959 − 0.281i)15-s + (−0.142 + 0.989i)16-s + (0.654 − 0.755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02389792403 - 0.7289611314i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02389792403 - 0.7289611314i\) |
\(L(1)\) |
\(\approx\) |
\(0.5095007080 - 0.5442675104i\) |
\(L(1)\) |
\(\approx\) |
\(0.5095007080 - 0.5442675104i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 + (-0.959 - 0.281i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 11 | \( 1 + (-0.415 - 0.909i)T \) |
| 13 | \( 1 + (-0.142 - 0.989i)T \) |
| 17 | \( 1 + (0.654 - 0.755i)T \) |
| 19 | \( 1 + (0.654 + 0.755i)T \) |
| 29 | \( 1 + (-0.654 + 0.755i)T \) |
| 31 | \( 1 + (-0.959 + 0.281i)T \) |
| 37 | \( 1 + (-0.841 - 0.540i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.142 - 0.989i)T \) |
| 59 | \( 1 + (-0.142 - 0.989i)T \) |
| 61 | \( 1 + (0.959 - 0.281i)T \) |
| 67 | \( 1 + (-0.415 + 0.909i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (-0.654 - 0.755i)T \) |
| 79 | \( 1 + (0.142 + 0.989i)T \) |
| 83 | \( 1 + (-0.841 - 0.540i)T \) |
| 89 | \( 1 + (0.959 + 0.281i)T \) |
| 97 | \( 1 + (-0.841 + 0.540i)T \) |
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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
−39.47384591982704763404433687334, −38.641699042349634593485111211132, −36.26019274866078620162480300100, −35.065145920863293422096012634568, −34.31946760242251160255153711669, −33.06883409596838763447412914249, −31.759518285674679575627498354396, −30.70849605596497712260666767523, −28.49575722422457959364713864996, −27.65446962190666171919289977351, −26.15187241659205154632258317873, −24.3959365379030146891103417381, −23.54723092607088623459520491724, −22.31273365780189005852132945833, −21.06397724226405307327047145086, −18.64106230986563546797929956735, −17.19886726270929591888650616212, −15.95332477594523285053167929874, −15.04640302322557239389925993895, −12.66888442624885074371064169073, −11.73938477192518769181868185940, −9.20888671721732226819136023697, −7.38276781385186051167280960646, −5.56424305609979977796163425944, −4.31223141471866953244943236567,
0.595428854419990019860432101880, 3.556840269818296224727192548207, 5.42494992536204878958907862173, 7.523772029316227977289894112619, 10.39764921993340918893402839025, 11.23050428241421999713945822326, 12.61097450498563826340730970543, 14.16792048252041723320680372312, 16.13280970468500293150500055710, 18.00311216792705612564796344821, 19.12407163549371921750645334304, 20.59192156171817513312413194609, 22.321976908452752798497500746831, 23.147002474980256875151377821040, 24.1372744651950129518469277798, 26.921519015905945165008564526051, 27.64332499310706950267022878752, 29.39296573621252220254676150618, 29.94165050014673494663967477452, 31.32782148765533013536235653519, 32.87170351617143714049148322345, 34.246312678623398647916604288300, 35.57374246302482518030052360961, 36.94920847979221336197334833455, 38.504239193860861151722615080878