L(s) = 1 | + i·5-s + (−1 − i)7-s + 11-s + (1 + i)17-s + i·19-s + (1 − i)23-s − 25-s + i·29-s + 31-s + (1 − i)35-s + 41-s + i·49-s + (−1 + i)53-s + i·55-s + i·59-s + ⋯ |
L(s) = 1 | + i·5-s + (−1 − i)7-s + 11-s + (1 + i)17-s + i·19-s + (1 − i)23-s − 25-s + i·29-s + 31-s + (1 − i)35-s + 41-s + i·49-s + (−1 + i)53-s + i·55-s + i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.083362744\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.083362744\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
good | 7 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−9.817285018251890730737094461439, −8.983044352910611235262376047170, −7.889104379218247201104466059728, −7.17467683447287300439154983343, −6.42501420608906908551344626600, −5.97066965238798052319730917113, −4.41326031949971868231633647385, −3.59793560300391206916986020241, −2.97790725315358273594864002799, −1.34082642338976775364110092032,
1.03116483750808352984009725253, 2.55699935156323982454638098293, 3.48768316540448623648721292838, 4.63850815168763443828917117720, 5.43908222485347726288412255275, 6.19403952810484311052916518592, 7.06408967374952688612976649064, 8.051854261980889296934999516978, 8.982538672279099593675159929244, 9.429301058329989485656193080175