L(s) = 1 | + (−0.215 − 0.976i)2-s + (0.798 − 0.601i)3-s + (−0.907 + 0.421i)4-s + (0.735 + 0.678i)5-s + (−0.759 − 0.650i)6-s + (−0.287 + 0.957i)7-s + (0.606 + 0.794i)8-s + (0.275 − 0.961i)9-s + (0.503 − 0.863i)10-s + (0.0558 + 0.998i)11-s + (−0.471 + 0.882i)12-s + (−0.982 − 0.185i)13-s + (0.997 + 0.0744i)14-s + (0.995 + 0.0991i)15-s + (0.645 − 0.763i)16-s + (−0.926 + 0.375i)17-s + ⋯ |
L(s) = 1 | + (−0.215 − 0.976i)2-s + (0.798 − 0.601i)3-s + (−0.907 + 0.421i)4-s + (0.735 + 0.678i)5-s + (−0.759 − 0.650i)6-s + (−0.287 + 0.957i)7-s + (0.606 + 0.794i)8-s + (0.275 − 0.961i)9-s + (0.503 − 0.863i)10-s + (0.0558 + 0.998i)11-s + (−0.471 + 0.882i)12-s + (−0.982 − 0.185i)13-s + (0.997 + 0.0744i)14-s + (0.995 + 0.0991i)15-s + (0.645 − 0.763i)16-s + (−0.926 + 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.431475818 + 0.008501260460i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431475818 + 0.008501260460i\) |
\(L(1)\) |
\(\approx\) |
\(1.132669851 - 0.2824782840i\) |
\(L(1)\) |
\(\approx\) |
\(1.132669851 - 0.2824782840i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.215 - 0.976i)T \) |
| 3 | \( 1 + (0.798 - 0.601i)T \) |
| 5 | \( 1 + (0.735 + 0.678i)T \) |
| 7 | \( 1 + (-0.287 + 0.957i)T \) |
| 11 | \( 1 + (0.0558 + 0.998i)T \) |
| 13 | \( 1 + (-0.982 - 0.185i)T \) |
| 17 | \( 1 + (-0.926 + 0.375i)T \) |
| 19 | \( 1 + (0.481 + 0.876i)T \) |
| 29 | \( 1 + (0.227 - 0.973i)T \) |
| 31 | \( 1 + (-0.166 + 0.985i)T \) |
| 37 | \( 1 + (0.130 + 0.991i)T \) |
| 41 | \( 1 + (0.813 - 0.581i)T \) |
| 43 | \( 1 + (0.299 + 0.954i)T \) |
| 47 | \( 1 + (0.854 + 0.519i)T \) |
| 53 | \( 1 + (0.503 + 0.863i)T \) |
| 59 | \( 1 + (0.879 - 0.476i)T \) |
| 61 | \( 1 + (0.0310 - 0.999i)T \) |
| 67 | \( 1 + (0.586 + 0.809i)T \) |
| 71 | \( 1 + (-0.191 + 0.981i)T \) |
| 73 | \( 1 + (0.227 + 0.973i)T \) |
| 79 | \( 1 + (-0.0186 - 0.999i)T \) |
| 83 | \( 1 + (-0.514 - 0.857i)T \) |
| 89 | \( 1 + (-0.999 - 0.0372i)T \) |
| 97 | \( 1 + (-0.944 - 0.329i)T \) |
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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
−24.079501679073147190219447580681, −22.46442077266323459998225326327, −21.95755109295446838560163650100, −21.02519528049894725437372622481, −19.85890404992781388454509562006, −19.57294902450551425830283403469, −18.23221257033824173445656892519, −17.283961330612716251700351634198, −16.5243580104810655369100815863, −16.07421519308904993814938671492, −14.98393759605962406990252523880, −13.976311192325139299359109813063, −13.64219006331443381650047371278, −12.8344356796983957538052176788, −10.98801274153726345072548875455, −10.05301015721231348235984962640, −9.26590477744213188223155053383, −8.76727640188035621652796383780, −7.621977589329934562372715084947, −6.81181218025659805015823442136, −5.52101834646256684520244359851, −4.7039379743676829601009159739, −3.84152876539420331813937994032, −2.40063060585998210650260370688, −0.75155522297631173016811490079,
1.60408088878623053603931197583, 2.392159965084171230410681171475, 2.933935288001278169309356088527, 4.25889628558640757609668014471, 5.61291211184523065448782603032, 6.79365475406942966946482015764, 7.77262926505301575571534281021, 8.83091615286040961277189030105, 9.66131388399103504256096944869, 10.129435540521631995144771125637, 11.52981555651974063648586337797, 12.4611742456625970754162374431, 12.911265055669260321455219339000, 14.027656426677728561614558498841, 14.64605548290694794362866875909, 15.53677735329074877021182063296, 17.3332401231019854581592361627, 17.79913521155291220811411920969, 18.61364293192595834257721359931, 19.235070662796973544252022378275, 20.05036832981993780966997960526, 20.86258540591314963629900085198, 21.75009824580895037610632866240, 22.37631338131115931644862605011, 23.20944325139436094771603112581