L(s) = 1 | + 5-s + 4.56·7-s − 2.56·11-s − 13-s − 2.56·17-s − 3.12·19-s − 6.56·23-s + 25-s + 1.12·29-s − 6·31-s + 4.56·35-s + 1.68·37-s − 0.561·41-s + 5.12·43-s − 2.87·47-s + 13.8·49-s + 7.68·53-s − 2.56·55-s − 12·59-s + 5.68·61-s − 65-s − 13.3·67-s − 14.5·71-s − 6·73-s − 11.6·77-s + 7.68·79-s − 16.4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.72·7-s − 0.772·11-s − 0.277·13-s − 0.621·17-s − 0.716·19-s − 1.36·23-s + 0.200·25-s + 0.208·29-s − 1.07·31-s + 0.771·35-s + 0.276·37-s − 0.0876·41-s + 0.781·43-s − 0.419·47-s + 1.97·49-s + 1.05·53-s − 0.345·55-s − 1.56·59-s + 0.727·61-s − 0.124·65-s − 1.63·67-s − 1.72·71-s − 0.702·73-s − 1.33·77-s + 0.864·79-s − 1.81·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 6.56T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 1.68T + 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 + 2.87T + 47T^{2} \) |
| 53 | \( 1 - 7.68T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 7.68T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 + 1.68T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.63745624210096357171202597080, −6.70668270789967633890177593551, −5.79986410952386088974612850144, −5.37785113468469007359117471042, −4.49960807842746904731067276608, −4.16076445854621581259954130651, −2.79710430694435538976692478721, −2.07720068936321360482182710078, −1.48507205216074216520246176703, 0,
1.48507205216074216520246176703, 2.07720068936321360482182710078, 2.79710430694435538976692478721, 4.16076445854621581259954130651, 4.49960807842746904731067276608, 5.37785113468469007359117471042, 5.79986410952386088974612850144, 6.70668270789967633890177593551, 7.63745624210096357171202597080