L(s) = 1 | + (−2.32 + 2.32i)3-s − 0.982i·7-s − 7.82i·9-s + (1.62 + 1.62i)11-s + (0.690 − 0.690i)13-s + 2.19·17-s + (−1.92 + 1.92i)19-s + (2.28 + 2.28i)21-s + 2.01i·23-s + (11.2 + 11.2i)27-s + (−5.27 + 5.27i)29-s − 0.435·31-s − 7.56·33-s + (5.79 + 5.79i)37-s + 3.21i·39-s + ⋯ |
L(s) = 1 | + (−1.34 + 1.34i)3-s − 0.371i·7-s − 2.60i·9-s + (0.490 + 0.490i)11-s + (0.191 − 0.191i)13-s + 0.532·17-s + (−0.441 + 0.441i)19-s + (0.498 + 0.498i)21-s + 0.420i·23-s + (2.15 + 2.15i)27-s + (−0.978 + 0.978i)29-s − 0.0781·31-s − 1.31·33-s + (0.953 + 0.953i)37-s + 0.514i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6088873993\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6088873993\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.32 - 2.32i)T - 3iT^{2} \) |
| 7 | \( 1 + 0.982iT - 7T^{2} \) |
| 11 | \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.690 + 0.690i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.19T + 17T^{2} \) |
| 19 | \( 1 + (1.92 - 1.92i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.01iT - 23T^{2} \) |
| 29 | \( 1 + (5.27 - 5.27i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.435T + 31T^{2} \) |
| 37 | \( 1 + (-5.79 - 5.79i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.93iT - 41T^{2} \) |
| 43 | \( 1 + (0.507 + 0.507i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.21T + 47T^{2} \) |
| 53 | \( 1 + (6.29 + 6.29i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.67 - 5.67i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.60 - 3.60i)T - 61iT^{2} \) |
| 67 | \( 1 + (-4.53 + 4.53i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 - 9.24iT - 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 + (0.683 - 0.683i)T - 83iT^{2} \) |
| 89 | \( 1 - 5.44iT - 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−9.826111087400369196589681074933, −9.409853985625967061455820433365, −8.326308177981786550527373819374, −7.14619853456450462743453301164, −6.34331409605662729333939444422, −5.60014319136424966059451572927, −4.82588586250590335237667402813, −4.05345269336132128909711948091, −3.31403295326408280917757148810, −1.27806931316978217599909850511,
0.31703204652175626910725345184, 1.51096480133520742044665413198, 2.54485177737414093906176184831, 4.12238028992274666891122400218, 5.23991348218712708713149098238, 5.94079204733758018017978448896, 6.46658743091078986498419670518, 7.31135225930243460666721856782, 8.011025591350059599235460900757, 8.892369681807902297449018878650