| L(s) = 1 | + (−1 − i)3-s + (1 − i)7-s − i·9-s + 4i·11-s + (−4 + 4i)13-s + (−4 − 4i)17-s + 4·19-s − 2·21-s + (5 + 5i)23-s + (−4 + 4i)27-s + 2i·29-s + 8i·31-s + (4 − 4i)33-s + 8·39-s − 4·41-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.577i)3-s + (0.377 − 0.377i)7-s − 0.333i·9-s + 1.20i·11-s + (−1.10 + 1.10i)13-s + (−0.970 − 0.970i)17-s + 0.917·19-s − 0.436·21-s + (1.04 + 1.04i)23-s + (−0.769 + 0.769i)27-s + 0.371i·29-s + 1.43i·31-s + (0.696 − 0.696i)33-s + 1.28·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9550078910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9550078910\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 + i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (4 + 4i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-5 - 5i)T + 23iT^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 8iT - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + (7 + 7i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3 + 3i)T - 47iT^{2} \) |
| 53 | \( 1 + (4 - 4i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 - 16iT - 71T^{2} \) |
| 73 | \( 1 + (-4 + 4i)T - 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (-5 - 5i)T + 83iT^{2} \) |
| 89 | \( 1 + 10iT - 89T^{2} \) |
| 97 | \( 1 + (-12 - 12i)T + 97iT^{2} \) |
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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
−9.435330880818362861898421988574, −8.977489032846631961492653431021, −7.54279911710181256328349125209, −6.99585954222057837778936537293, −6.74290309768572867376203147843, −5.15012851878892312761354036258, −4.88516503467094398560240983087, −3.62600451083269632399138749568, −2.26110268668030789283699228157, −1.21720050887824962521273557664,
0.42983496778959672616558086028, 2.22956301135360237128084143282, 3.23976030722810859147189041380, 4.46462991134290027235382930180, 5.19295934726610047977433885556, 5.79225364350964496423268846699, 6.73297676475925844216356226049, 7.970434541080482739222639776938, 8.310488556918893265567780529407, 9.403435773382798172488660615074
