L(s) = 1 | + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.342 + 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.642 + 0.766i)21-s + (0.173 + 0.984i)23-s + (0.939 + 0.342i)25-s + (0.866 + 0.5i)27-s + (−0.642 − 0.766i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)3-s + (−0.984 − 0.173i)5-s + (−0.5 − 0.866i)7-s + (−0.766 + 0.642i)9-s + (−0.866 − 0.5i)11-s + (−0.342 + 0.939i)13-s + (0.173 + 0.984i)15-s + (0.766 + 0.642i)17-s + (−0.642 + 0.766i)21-s + (0.173 + 0.984i)23-s + (0.939 + 0.342i)25-s + (0.866 + 0.5i)27-s + (−0.642 − 0.766i)29-s + (−0.5 − 0.866i)31-s + (−0.173 + 0.984i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0976 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0976 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09207824576 + 0.1015502713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09207824576 + 0.1015502713i\) |
\(L(1)\) |
\(\approx\) |
\(0.5322664584 - 0.1620829706i\) |
\(L(1)\) |
\(\approx\) |
\(0.5322664584 - 0.1620829706i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.342 - 0.939i)T \) |
| 5 | \( 1 + (-0.984 - 0.173i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 + (-0.342 + 0.939i)T \) |
| 17 | \( 1 + (0.766 + 0.642i)T \) |
| 23 | \( 1 + (0.173 + 0.984i)T \) |
| 29 | \( 1 + (-0.642 - 0.766i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.939 + 0.342i)T \) |
| 43 | \( 1 + (0.984 + 0.173i)T \) |
| 47 | \( 1 + (-0.766 + 0.642i)T \) |
| 53 | \( 1 + (-0.984 + 0.173i)T \) |
| 59 | \( 1 + (0.642 - 0.766i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.642 + 0.766i)T \) |
| 71 | \( 1 + (-0.173 + 0.984i)T \) |
| 73 | \( 1 + (0.939 - 0.342i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.766 - 0.642i)T \) |
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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
−25.28549717394390894703670023599, −24.03488387255045789154003356324, −22.86108874213666275157513737320, −22.68775637031553186889067557077, −21.5700649131094375522677648262, −20.58455424024864116560558754139, −19.86173343189764075358257828722, −18.66947533669080840994995713289, −17.94789604567764702117944291900, −16.55844025346090563753035066708, −15.89679231550014193255531420654, −15.20397401144815703730460534874, −14.49400949893380210952077516889, −12.65337140967113986117055910203, −12.18927267531508658325824777708, −10.98027693811074728694336038780, −10.228633501105170703999894004633, −9.16764149321207191124069442438, −8.13675905613225719446692962976, −6.99866049114126914352059811553, −5.54924591323976789110657071806, −4.88455399573157039185800103193, −3.49290600819255762614138845192, −2.7309024682360577304298954273, −0.0966646774812105548995936634,
1.366751807121300959245263866373, 3.00691921985214992572387022914, 4.14610317295147394939341123930, 5.47253650959471801374105345805, 6.65898981962418651196034995998, 7.59063065343163715098485830695, 8.15685105920652902819690826763, 9.66841233029156116804363847415, 10.97077480603458034597454678537, 11.62432890541623021728183803785, 12.71258589995090886294341543325, 13.37633946208792811640993135243, 14.40751918276164109814130768874, 15.6914607109144564080071246787, 16.65836628671543726592347123359, 17.187716156195312573950530918049, 18.70446825660106552073703309959, 19.083066080077577445700985344387, 19.91637242435449008841173288369, 20.94538898885111152884026016534, 22.243143328078746073785574592439, 23.20624530309660206613816319520, 23.78992571637374106157817066445, 24.22167345173606287875529801021, 25.6641831909426760270182434881