L(s) = 1 | + i·3-s + i·5-s + 3.34·7-s − 9-s − 3.80·11-s − 4.01·13-s − 15-s + 3.43i·17-s − 2.65·19-s + 3.34i·21-s + (−4.71 + 0.853i)23-s − 25-s − i·27-s + 2.73·29-s + 5.29i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 1.26·7-s − 0.333·9-s − 1.14·11-s − 1.11·13-s − 0.258·15-s + 0.833i·17-s − 0.608·19-s + 0.730i·21-s + (−0.984 + 0.178i)23-s − 0.200·25-s − 0.192i·27-s + 0.507·29-s + 0.950i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1872822150\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1872822150\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 23 | \( 1 + (4.71 - 0.853i)T \) |
good | 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 - 3.43iT - 17T^{2} \) |
| 19 | \( 1 + 2.65T + 19T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 - 5.29iT - 31T^{2} \) |
| 37 | \( 1 + 1.26iT - 37T^{2} \) |
| 41 | \( 1 + 0.928T + 41T^{2} \) |
| 43 | \( 1 - 6.49T + 43T^{2} \) |
| 47 | \( 1 + 1.22iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 15.1iT - 59T^{2} \) |
| 61 | \( 1 + 5.10iT - 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 4.10iT - 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 0.329iT - 89T^{2} \) |
| 97 | \( 1 + 4.88iT - 97T^{2} \) |
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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
−8.140462520331708489014868837885, −7.37117869967805063284375868715, −6.49890252431191221211106462577, −5.57468384660161187521598320152, −4.98331480291268165090575705477, −4.39848776074109126419983470276, −3.46679070709519232263965257721, −2.45631144113060708518642194587, −1.81686279643437763936853894460, −0.04757360035479691962131988706,
1.14446815940104791787642522994, 2.28404599212262224311970332760, 2.65967999849021894802153316769, 4.24010177872993294921628000089, 4.73801201544707558230008705679, 5.46249600986723271174085727067, 6.09950962610524829698790624593, 7.36633097736439583126383489460, 7.53135359258042348732008816915, 8.266588036864386407911591027185