| L(s) = 1 | − 1.89·5-s + 1.71·7-s + 0.839·11-s − 1.94·13-s − 4.19·17-s + 1.07·19-s − 0.340·23-s − 1.42·25-s − 4.65·29-s + 9.22·31-s − 3.23·35-s − 3.13·37-s − 5.59·41-s + 1.01·43-s − 10.4·47-s − 4.07·49-s + 9.10·53-s − 1.58·55-s + 59-s − 0.592·61-s + 3.68·65-s − 1.37·67-s − 11.2·71-s + 8.50·73-s + 1.43·77-s − 11.4·79-s − 3.28·83-s + ⋯ |
| L(s) = 1 | − 0.845·5-s + 0.646·7-s + 0.253·11-s − 0.540·13-s − 1.01·17-s + 0.245·19-s − 0.0710·23-s − 0.284·25-s − 0.865·29-s + 1.65·31-s − 0.547·35-s − 0.515·37-s − 0.874·41-s + 0.154·43-s − 1.53·47-s − 0.581·49-s + 1.25·53-s − 0.214·55-s + 0.130·59-s − 0.0758·61-s + 0.457·65-s − 0.167·67-s − 1.33·71-s + 0.995·73-s + 0.163·77-s − 1.28·79-s − 0.360·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.89T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 0.839T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 0.340T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 9.22T + 31T^{2} \) |
| 37 | \( 1 + 3.13T + 37T^{2} \) |
| 41 | \( 1 + 5.59T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.10T + 53T^{2} \) |
| 61 | \( 1 + 0.592T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 8.50T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.505817524363088707489358993676, −8.033238884551797136832861646035, −7.19427569631828559564405074351, −6.50987751350535270782008287168, −5.37043659193555578445913184308, −4.56997464048030526651031023695, −3.88976186730296706157602217288, −2.76309573707667528840644904144, −1.59161610485855730457749546170, 0,
1.59161610485855730457749546170, 2.76309573707667528840644904144, 3.88976186730296706157602217288, 4.56997464048030526651031023695, 5.37043659193555578445913184308, 6.50987751350535270782008287168, 7.19427569631828559564405074351, 8.033238884551797136832861646035, 8.505817524363088707489358993676
