L(s) = 1 | + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯ |
L(s) = 1 | + (0.945 + 0.327i)2-s + (−0.992 − 0.118i)3-s + (0.786 + 0.618i)4-s + (−0.841 + 0.540i)5-s + (−0.899 − 0.436i)6-s + (−0.739 − 0.672i)7-s + (0.540 + 0.841i)8-s + (0.971 + 0.235i)9-s + (−0.971 + 0.235i)10-s + (−0.998 − 0.0475i)11-s + (−0.707 − 0.707i)12-s + (−0.479 − 0.877i)14-s + (0.899 − 0.436i)15-s + (0.235 + 0.971i)16-s + (0.945 − 0.327i)17-s + (0.841 + 0.540i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.058826109 + 0.8626787433i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.058826109 + 0.8626787433i\) |
\(L(1)\) |
\(\approx\) |
\(1.060382902 + 0.3281002111i\) |
\(L(1)\) |
\(\approx\) |
\(1.060382902 + 0.3281002111i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (0.945 + 0.327i)T \) |
| 3 | \( 1 + (-0.992 - 0.118i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.739 - 0.672i)T \) |
| 11 | \( 1 + (-0.998 - 0.0475i)T \) |
| 17 | \( 1 + (0.945 - 0.327i)T \) |
| 19 | \( 1 + (0.853 - 0.520i)T \) |
| 23 | \( 1 + (-0.0237 - 0.999i)T \) |
| 29 | \( 1 + (0.952 - 0.304i)T \) |
| 31 | \( 1 + (0.479 + 0.877i)T \) |
| 37 | \( 1 + (-0.258 + 0.965i)T \) |
| 41 | \( 1 + (-0.393 + 0.919i)T \) |
| 43 | \( 1 + (-0.952 - 0.304i)T \) |
| 47 | \( 1 + (-0.142 + 0.989i)T \) |
| 53 | \( 1 + (-0.989 + 0.142i)T \) |
| 59 | \( 1 + (0.118 + 0.992i)T \) |
| 61 | \( 1 + (0.636 + 0.771i)T \) |
| 67 | \( 1 + (0.618 + 0.786i)T \) |
| 71 | \( 1 + (0.888 - 0.458i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (0.281 - 0.959i)T \) |
| 83 | \( 1 + (-0.0713 - 0.997i)T \) |
| 97 | \( 1 + (0.998 - 0.0475i)T \) |
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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
−21.232039981212211123794084032296, −20.60416086417323722837258033865, −19.58722388261036224564078437597, −18.91396204851552759508961250884, −18.249222850360062977237374487673, −16.938043304229475482015294453345, −16.19604278429558720174367450102, −15.661746672224080065447285445895, −15.236752315936518838547529431281, −13.93025963756153317868351461474, −12.90048693417828753393239434236, −12.4775259210888881530271993889, −11.84799665060010949625718309, −11.18484778729803421592592245282, −10.14864187973291535872223048927, −9.60502265538406108086500377562, −8.09898713892463596578531583618, −7.26386852330506840300823476895, −6.28029142688038174250655078734, −5.3199791333905926013941018706, −5.11916685019288647025764174383, −3.79525424242472350809823581855, −3.24655180293838779866580239740, −1.83645556001386922124948680701, −0.58358094122315325465707413043,
0.95430948072527179934408867882, 2.77246067701670815139487230066, 3.34901961501024115907842649888, 4.50875403076738504531118072861, 5.04510184406324182283732958658, 6.191817407467159312433208288710, 6.81743037229830291874587007527, 7.47715499983572141332961135271, 8.214080778817705837648018430301, 10.096473687408083541046777100185, 10.48712651431238283687895522409, 11.47479733687913400140326744840, 12.04740947143120848715909495738, 12.81808739909494279675601266536, 13.55695403814368723767499116654, 14.396203651112319440238608166383, 15.49451854192617418411133944979, 16.00760189358199232583471525130, 16.461124603674309220656099337276, 17.41611274850887567673089651647, 18.352940953662762924437831389246, 19.10159071324433301796265326792, 20.03961561269540925619913862001, 20.821905692870331341537920281474, 21.751439076707874305469403445627