L(s) = 1 | + (1.37 − 0.346i)2-s + (−1.70 + 0.293i)3-s + (1.75 − 0.950i)4-s + 2.20i·5-s + (−2.23 + 0.993i)6-s + 3.71i·7-s + (2.08 − 1.91i)8-s + (2.82 − 1.00i)9-s + (0.762 + 3.01i)10-s − 0.958·11-s + (−2.72 + 2.13i)12-s + 2.52·13-s + (1.28 + 5.10i)14-s + (−0.645 − 3.75i)15-s + (2.19 − 3.34i)16-s + 2.72i·17-s + ⋯ |
L(s) = 1 | + (0.969 − 0.245i)2-s + (−0.985 + 0.169i)3-s + (0.879 − 0.475i)4-s + 0.984i·5-s + (−0.914 + 0.405i)6-s + 1.40i·7-s + (0.736 − 0.676i)8-s + (0.942 − 0.333i)9-s + (0.241 + 0.954i)10-s − 0.288·11-s + (−0.786 + 0.617i)12-s + 0.701·13-s + (0.344 + 1.36i)14-s + (−0.166 − 0.969i)15-s + (0.548 − 0.836i)16-s + 0.660i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62605 + 0.561674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62605 + 0.561674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 + 0.346i)T \) |
| 3 | \( 1 + (1.70 - 0.293i)T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 2.20iT - 5T^{2} \) |
| 7 | \( 1 - 3.71iT - 7T^{2} \) |
| 11 | \( 1 + 0.958T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 - 2.72iT - 17T^{2} \) |
| 19 | \( 1 + 0.627iT - 19T^{2} \) |
| 29 | \( 1 + 6.59iT - 29T^{2} \) |
| 31 | \( 1 + 9.19iT - 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 5.91iT - 41T^{2} \) |
| 43 | \( 1 - 7.53iT - 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 0.0894T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 4.66T + 73T^{2} \) |
| 79 | \( 1 + 6.77iT - 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 0.323iT - 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.89525658133844619035266681288, −11.23557739361599094907199963047, −10.52641690492253527544604885922, −9.506534602338622390867083326882, −7.83682813534002201881796857707, −6.37870017979934830963031425577, −6.06797349434158263551110051292, −4.94141198279328607676820606938, −3.56865794360735247353710143420, −2.19167931202432125315529447775,
1.27225596645826843074865580371, 3.70552455607667084537976431205, 4.75468423621503893445040592818, 5.47077109811388574970646472653, 6.79564125394008762283992443895, 7.41954919728952615445863552475, 8.737180665836742355561087579837, 10.48618467686097694935168317089, 10.83589601921442357485651438785, 12.15949694368517143399747857312