| L(s) = 1 | − i·2-s − 4-s + (−2 − i)5-s + i·7-s + i·8-s + (−1 + 2i)10-s − 4·11-s − 6i·13-s + 14-s + 16-s − 4i·17-s + (2 + i)20-s + 4i·22-s + 4i·23-s + (3 + 4i)25-s − 6·26-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.377i·7-s + 0.353i·8-s + (−0.316 + 0.632i)10-s − 1.20·11-s − 1.66i·13-s + 0.267·14-s + 0.250·16-s − 0.970i·17-s + (0.447 + 0.223i)20-s + 0.852i·22-s + 0.834i·23-s + (0.600 + 0.800i)25-s − 1.17·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3456003557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3456003557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 13iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.469463338640619199687141540940, −8.497004317098699165190103564565, −7.910722005995596124565416693415, −7.38012966793484340496372093760, −5.85631213248439403890025949048, −5.15915451911064044933536541852, −4.47193562158291097772231791017, −3.17439252291440844827159302477, −2.76297842385605601490845328848, −1.07333505384317772197202160200,
0.14666739278379121765149094181, 2.05925313190700804681672203670, 3.40334233731209527438055502086, 4.23418191196290084247982383671, 4.91890386073066649669256261273, 6.07955646815413868287928711214, 6.90369682292706166472455160857, 7.33180364803405527738611823975, 8.416117615896282663671511083726, 8.594913676883796212663754559519
