L(s) = 1 | − i·2-s + 2.76i·3-s − 4-s + (−2.19 + 0.432i)5-s + 2.76·6-s + 0.761i·7-s + i·8-s − 4.62·9-s + (0.432 + 2.19i)10-s − 0.864·11-s − 2.76i·12-s + 5.62i·13-s + 0.761·14-s + (−1.19 − 6.05i)15-s + 16-s + 3.62i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.59i·3-s − 0.5·4-s + (−0.981 + 0.193i)5-s + 1.12·6-s + 0.287i·7-s + 0.353i·8-s − 1.54·9-s + (0.136 + 0.693i)10-s − 0.260·11-s − 0.797i·12-s + 1.56i·13-s + 0.203·14-s + (−0.308 − 1.56i)15-s + 0.250·16-s + 0.879i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.511151 + 0.621707i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.511151 + 0.621707i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 + (2.19 - 0.432i)T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2.76iT - 3T^{2} \) |
| 7 | \( 1 - 0.761iT - 7T^{2} \) |
| 11 | \( 1 + 0.864T + 11T^{2} \) |
| 13 | \( 1 - 5.62iT - 13T^{2} \) |
| 17 | \( 1 - 3.62iT - 17T^{2} \) |
| 23 | \( 1 + 8.01iT - 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 + 0.476iT - 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 6.86iT - 43T^{2} \) |
| 47 | \( 1 + 1.25iT - 47T^{2} \) |
| 53 | \( 1 - 2.37iT - 53T^{2} \) |
| 59 | \( 1 + 4.49T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 1.03iT - 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 16.4iT - 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 0.270iT - 83T^{2} \) |
| 89 | \( 1 + 0.387T + 89T^{2} \) |
| 97 | \( 1 - 8.50iT - 97T^{2} \) |
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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
−12.41077713380571947027601922886, −11.67293741349906296720479084870, −10.75375989109082026035379010027, −10.14884691040259805160011263335, −8.966339076742160409515152000891, −8.305784216425290768458823111054, −6.43802464655902387947760643199, −4.61589358589630215739542468215, −4.22455135236498366305667709228, −2.85465039987396600095188827587,
0.74776098080473919257872564369, 3.12557198292951037263457555753, 4.98418674247858018663533438450, 6.24733095303766015821591575478, 7.44196323962049808486462338607, 7.75496749153089660814551954387, 8.715826399655238284055447667960, 10.36118109238719800379241836569, 11.74260368640394501588474954878, 12.37068294356348992738179152336