L(s) = 1 | + (0.580 + 0.814i)2-s + (−0.235 − 0.971i)3-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (0.995 + 0.0950i)12-s + (0.142 + 0.989i)13-s + (−0.959 + 0.281i)15-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (−0.888 − 0.458i)18-s + (−0.981 + 0.189i)19-s + ⋯ |
L(s) = 1 | + (0.580 + 0.814i)2-s + (−0.235 − 0.971i)3-s + (−0.327 + 0.945i)4-s + (−0.0475 − 0.998i)5-s + (0.654 − 0.755i)6-s + (−0.959 + 0.281i)8-s + (−0.888 + 0.458i)9-s + (0.786 − 0.618i)10-s + (0.580 − 0.814i)11-s + (0.995 + 0.0950i)12-s + (0.142 + 0.989i)13-s + (−0.959 + 0.281i)15-s + (−0.786 − 0.618i)16-s + (−0.981 − 0.189i)17-s + (−0.888 − 0.458i)18-s + (−0.981 + 0.189i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05799710300 - 0.3083223169i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05799710300 - 0.3083223169i\) |
\(L(1)\) |
\(\approx\) |
\(0.9087416500 + 4.633020717\times10^{-5}i\) |
\(L(1)\) |
\(\approx\) |
\(0.9087416500 + 4.633020717\times10^{-5}i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.580 + 0.814i)T \) |
| 3 | \( 1 + (-0.235 - 0.971i)T \) |
| 5 | \( 1 + (-0.0475 - 0.998i)T \) |
| 11 | \( 1 + (0.580 - 0.814i)T \) |
| 13 | \( 1 + (0.142 + 0.989i)T \) |
| 17 | \( 1 + (-0.981 - 0.189i)T \) |
| 19 | \( 1 + (-0.981 + 0.189i)T \) |
| 29 | \( 1 + (-0.654 + 0.755i)T \) |
| 31 | \( 1 + (-0.723 - 0.690i)T \) |
| 37 | \( 1 + (-0.888 + 0.458i)T \) |
| 41 | \( 1 + (-0.841 + 0.540i)T \) |
| 43 | \( 1 + (-0.959 - 0.281i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.928 - 0.371i)T \) |
| 59 | \( 1 + (0.786 - 0.618i)T \) |
| 61 | \( 1 + (-0.235 + 0.971i)T \) |
| 67 | \( 1 + (-0.995 + 0.0950i)T \) |
| 71 | \( 1 + (0.415 - 0.909i)T \) |
| 73 | \( 1 + (0.327 - 0.945i)T \) |
| 79 | \( 1 + (0.928 + 0.371i)T \) |
| 83 | \( 1 + (-0.841 - 0.540i)T \) |
| 89 | \( 1 + (-0.723 + 0.690i)T \) |
| 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
−27.918126814905210050214910231524, −27.31358031041326792258444785799, −26.27266986965110683750786503184, −25.16002580696750096131478867172, −23.57885573590993911448225505990, −22.67110754703729802424369073294, −22.24669277537523780677269891767, −21.292444482095222071950626452561, −20.2365188151549250597019438379, −19.486276813618444208018836647749, −18.12213438968588198545672467530, −17.282549498883707484863632345922, −15.42196868209434490231612634651, −15.090581048325853825798916007296, −14.05117006650456391727584207862, −12.71431371079358674979297944620, −11.50493352314657754914554661434, −10.691921897082617076610610203492, −9.99717980237771323664735793204, −8.80373773750511412641762959318, −6.78357577352587783488893963135, −5.6639649621908525418392957233, −4.361978962828120547859216239946, −3.46275416517573448314583066015, −2.203232617761227414567510400852,
0.09186006461360882276648228828, 1.88438608153628272586554823873, 3.80705594040924683300640337302, 5.07028581738440689303316512628, 6.18650551667205287639370757270, 7.05042118985681355785900319486, 8.450539926535296915864512126048, 8.9459414478989949780639308998, 11.34704889840550724027998393309, 12.1434150871390619631723192507, 13.218715140793596626506932388583, 13.7642524273296828835516634066, 15.03070735343616493021840495567, 16.63270514890484720254028345026, 16.72121530121950322940805303182, 18.01499936543409078016222370963, 19.12871678044855851645861162438, 20.28586617606925572709963508208, 21.51069708269162693981696601851, 22.43631354123575222089365424346, 23.773963477697609166286718854414, 24.010379430076695771987477603771, 24.91862572511620814998786676634, 25.748830946996553925839899559363, 26.999783740353581336496197807341