L(s) = 1 | + (−1.15 − 1.15i)3-s − 4.31·7-s − 0.316i·9-s + (0.158 + 0.158i)11-s + (−2.31 − 2.31i)13-s − 5.31i·17-s + (3.15 − 3.15i)19-s + (5 + 5i)21-s + 6.31·23-s + (−3.84 + 3.84i)27-s + (−2 + 2i)29-s − 4.31·31-s − 0.366i·33-s + (−7.31 + 7.31i)37-s + 5.36i·39-s + ⋯ |
L(s) = 1 | + (−0.668 − 0.668i)3-s − 1.63·7-s − 0.105i·9-s + (0.0477 + 0.0477i)11-s + (−0.642 − 0.642i)13-s − 1.28i·17-s + (0.724 − 0.724i)19-s + (1.09 + 1.09i)21-s + 1.31·23-s + (−0.739 + 0.739i)27-s + (−0.371 + 0.371i)29-s − 0.775·31-s − 0.0638i·33-s + (−1.20 + 1.20i)37-s + 0.859i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0708 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1353939183\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1353939183\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (1.15 + 1.15i)T + 3iT^{2} \) |
| 7 | \( 1 + 4.31T + 7T^{2} \) |
| 11 | \( 1 + (-0.158 - 0.158i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.31 + 2.31i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.31iT - 17T^{2} \) |
| 19 | \( 1 + (-3.15 + 3.15i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 + (2 - 2i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.31T + 31T^{2} \) |
| 37 | \( 1 + (7.31 - 7.31i)T - 37iT^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + (5.63 - 5.63i)T - 43iT^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + (-3.31 + 3.31i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.31 - 5.31i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.63 - 3.63i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.84 + 5.84i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.63iT - 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 + (3.84 + 3.84i)T + 83iT^{2} \) |
| 89 | \( 1 - 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 6.63iT - 97T^{2} \) |
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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
−9.394268313623513324547937699939, −9.226497596604096953584325243439, −7.73906055003847154274142522199, −6.84148625800762159590602136275, −6.74020335000088255489408589652, −5.57290568211868729647569139545, −4.92562980732840772962758132746, −3.33630793345540632330693706997, −2.84475279641438534551156746270, −1.02070425431454054163475831859,
0.06740925567190375817104291589, 2.04172371008821753623442869207, 3.43459648962968970039992280655, 4.00107790770485274444010507912, 5.26091487679869548973800404424, 5.77308144390051803854184154127, 6.76217304256824510487475551852, 7.36104737819671467836863527549, 8.649382187336308471161321630448, 9.367267724133466789648100182779