L(s) = 1 | − 2-s + i·3-s + 4-s − i·6-s − 3i·7-s − 8-s − 9-s − 5i·11-s + i·12-s + 6·13-s + 3i·14-s + 16-s + (−4 + i)17-s + 18-s + 5·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.408i·6-s − 1.13i·7-s − 0.353·8-s − 0.333·9-s − 1.50i·11-s + 0.288i·12-s + 1.66·13-s + 0.801i·14-s + 0.250·16-s + (−0.970 + 0.242i)17-s + 0.235·18-s + 1.14·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.132082520\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132082520\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (4 - i)T \) |
good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 + 5iT - 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 10iT - 41T^{2} \) |
| 43 | \( 1 + 9T + 43T^{2} \) |
| 47 | \( 1 - 7T + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10iT - 61T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + 10iT - 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.750097111912997300861064047696, −8.171950665363564383955980946475, −7.36494255444038018413804892445, −6.37423533405022419345702480257, −5.86812092560260338695919931056, −4.72409965906794031891673785284, −3.54814187292834819690867394929, −3.32781846637359992257630067843, −1.55356393873671629363159143849, −0.52213276432786156045869401896,
1.26208608543052845014754216395, 2.15001075824131315984038297081, 2.99786158009943644466650984735, 4.29941942992926682234537392217, 5.37775793615889047836584081626, 6.15829847009037121729998324506, 6.85995922850034180564809677020, 7.55534081073156664536225998046, 8.460748441561197525015952767386, 8.939356424860346497372138198987