L(s) = 1 | + i·3-s + 5i·7-s − 9-s − 6·11-s − 3i·13-s + 2i·17-s − 19-s − 5·21-s − 2i·23-s − i·27-s − 6·29-s + 3·31-s − 6i·33-s + 6i·37-s + 3·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.88i·7-s − 0.333·9-s − 1.80·11-s − 0.832i·13-s + 0.485i·17-s − 0.229·19-s − 1.09·21-s − 0.417i·23-s − 0.192i·27-s − 1.11·29-s + 0.538·31-s − 1.04i·33-s + 0.986i·37-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.195881 + 0.829769i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.195881 + 0.829769i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 11iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.95925970321933935036357348622, −10.15006075759147659366982183866, −9.330922705046043833938664371728, −8.345507522572148688284547905203, −7.895026211609946857742827648042, −6.17774902954367476665381116904, −5.50400654566528170230515282740, −4.77633521378316265655297466110, −3.08745892033247913453334961768, −2.37240909098392244819698094744,
0.44493886281827968706292317175, 2.10768280699253065414019768653, 3.55985029952620517947321345456, 4.63144159076330806147000252771, 5.72923764441056247193135998389, 7.16363970047527631201995177066, 7.31710579453034447422737106350, 8.305381023859202545485121613592, 9.537490635967879284215278258656, 10.51100617914757906596746223324