| L(s) = 1 | + (−2.15 − 2.15i)3-s − 2.31·7-s + 6.31i·9-s + (−3.15 − 3.15i)11-s + (−4.31 − 4.31i)13-s − 1.31i·17-s + (−0.158 + 0.158i)19-s + (5 + 5i)21-s + 0.316·23-s + (7.15 − 7.15i)27-s + (−2 + 2i)29-s + 2.31·31-s + 13.6i·33-s + (0.683 − 0.683i)37-s + 18.6i·39-s + ⋯ |
| L(s) = 1 | + (−1.24 − 1.24i)3-s − 0.875·7-s + 2.10i·9-s + (−0.952 − 0.952i)11-s + (−1.19 − 1.19i)13-s − 0.319i·17-s + (−0.0363 + 0.0363i)19-s + (1.09 + 1.09i)21-s + 0.0660·23-s + (1.37 − 1.37i)27-s + (−0.371 + 0.371i)29-s + 0.416·31-s + 2.37i·33-s + (0.112 − 0.112i)37-s + 2.98i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1425643733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1425643733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.15 + 2.15i)T + 3iT^{2} \) |
| 7 | \( 1 + 2.31T + 7T^{2} \) |
| 11 | \( 1 + (3.15 + 3.15i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.31 + 4.31i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.31iT - 17T^{2} \) |
| 19 | \( 1 + (0.158 - 0.158i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.316T + 23T^{2} \) |
| 29 | \( 1 + (2 - 2i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + (-0.683 + 0.683i)T - 37iT^{2} \) |
| 41 | \( 1 - 5iT - 41T^{2} \) |
| 43 | \( 1 + (7.63 - 7.63i)T - 43iT^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + (-3.31 + 3.31i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.31 + 1.31i)T + 59iT^{2} \) |
| 61 | \( 1 + (-9.63 + 9.63i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.15 - 9.15i)T + 67iT^{2} \) |
| 71 | \( 1 - 8.63iT - 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 - 4.31T + 79T^{2} \) |
| 83 | \( 1 + (-7.15 - 7.15i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.94iT - 89T^{2} \) |
| 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
−9.798775821807177050906156483987, −8.350922380817250810424892825797, −7.77674752689003720997017539214, −6.96160185690865698015753500902, −6.32249683890767939218438704300, −5.43596447369959346599853739672, −5.05196913864700507464798102827, −3.25104307527819371630903866192, −2.36926723439860301322166052137, −0.76566600049742671456177411934,
0.093363555679981067424074398872, 2.27146499289248078884599076887, 3.59204921142588349302975206673, 4.53121800582566199119998475308, 5.01820908323059076375563169390, 5.95408983375833606487269765453, 6.74578221472860655166256346426, 7.48250433491511633635773815926, 8.897831630170076456882741381280, 9.690395694985567966652451958250
