L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s − 28-s + 29-s − 31-s + 32-s + 34-s + 35-s + 37-s + 38-s − 40-s − 41-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 20-s + 22-s − 23-s + 25-s − 26-s − 28-s + 29-s − 31-s + 32-s + 34-s + 35-s + 37-s + 38-s − 40-s − 41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.381123394\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.381123394\) |
\(L(1)\) |
\(\approx\) |
\(1.654482656\) |
\(L(1)\) |
\(\approx\) |
\(1.654482656\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 379 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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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
−21.72170515008926437471657270086, −20.25648269509997242140301688220, −19.98673599551782539642203350179, −19.32972891685378211407120105236, −18.508867715864070820992831757975, −17.074771846169276633592429580166, −16.39035491353956784002609582699, −15.919230528844772556170582114528, −14.93990341648669049893803916282, −14.40578668272472399912048148424, −13.53055278995011980767878745383, −12.47636804693897536229747820770, −12.04978071869979681292172495262, −11.50028274006675727584629046602, −10.244073589237651030284886315399, −9.62626723256881043463976592705, −8.31319944501526083283692015840, −7.33426016980424238057786456243, −6.84296089877557630559711796857, −5.82815593146849663013871201893, −4.91813877151357170125031514386, −3.853240093842573386855073908101, −3.43388739870701150989044302234, −2.41856541551362287129879271624, −0.94793428880841682403797747848,
0.94793428880841682403797747848, 2.41856541551362287129879271624, 3.43388739870701150989044302234, 3.853240093842573386855073908101, 4.91813877151357170125031514386, 5.82815593146849663013871201893, 6.84296089877557630559711796857, 7.33426016980424238057786456243, 8.31319944501526083283692015840, 9.62626723256881043463976592705, 10.244073589237651030284886315399, 11.50028274006675727584629046602, 12.04978071869979681292172495262, 12.47636804693897536229747820770, 13.53055278995011980767878745383, 14.40578668272472399912048148424, 14.93990341648669049893803916282, 15.919230528844772556170582114528, 16.39035491353956784002609582699, 17.074771846169276633592429580166, 18.508867715864070820992831757975, 19.32972891685378211407120105236, 19.98673599551782539642203350179, 20.25648269509997242140301688220, 21.72170515008926437471657270086