| L(s) = 1 | + 2.30·2-s + 3.33·4-s + 3.03·5-s − 4.39·7-s + 3.07·8-s + 7.01·10-s − 5.68·11-s + 3.95·13-s − 10.1·14-s + 0.441·16-s − 3.21·17-s − 3.61·19-s + 10.1·20-s − 13.1·22-s + 2.69·23-s + 4.21·25-s + 9.12·26-s − 14.6·28-s − 0.823·31-s − 5.13·32-s − 7.42·34-s − 13.3·35-s − 5.60·37-s − 8.35·38-s + 9.34·40-s + 0.558·41-s − 12.7·43-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 1.66·4-s + 1.35·5-s − 1.66·7-s + 1.08·8-s + 2.21·10-s − 1.71·11-s + 1.09·13-s − 2.71·14-s + 0.110·16-s − 0.780·17-s − 0.830·19-s + 2.26·20-s − 2.79·22-s + 0.562·23-s + 0.843·25-s + 1.78·26-s − 2.76·28-s − 0.147·31-s − 0.907·32-s − 1.27·34-s − 2.25·35-s − 0.921·37-s − 1.35·38-s + 1.47·40-s + 0.0872·41-s − 1.93·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 2.30T + 2T^{2} \) |
| 5 | \( 1 - 3.03T + 5T^{2} \) |
| 7 | \( 1 + 4.39T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 2.69T + 23T^{2} \) |
| 31 | \( 1 + 0.823T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 0.558T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 0.129T + 47T^{2} \) |
| 53 | \( 1 + 6.88T + 53T^{2} \) |
| 59 | \( 1 - 0.745T + 59T^{2} \) |
| 61 | \( 1 + 7.17T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 8.81T + 73T^{2} \) |
| 79 | \( 1 + 6.99T + 79T^{2} \) |
| 83 | \( 1 + 8.90T + 83T^{2} \) |
| 89 | \( 1 - 1.51T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−6.94461548148676587537001627190, −6.52148776472098078457342259762, −6.07972822745868391255647484318, −5.40197051930943164168557299651, −4.91631343309763171596858957657, −3.84501330358314459597575868574, −3.15719808168103440961908448365, −2.58303250537729844304075052714, −1.84718616729897050570614908521, 0,
1.84718616729897050570614908521, 2.58303250537729844304075052714, 3.15719808168103440961908448365, 3.84501330358314459597575868574, 4.91631343309763171596858957657, 5.40197051930943164168557299651, 6.07972822745868391255647484318, 6.52148776472098078457342259762, 6.94461548148676587537001627190
