| L(s) = 1 | + 3.60·5-s + 3.60·11-s − 6·13-s + 7.21·17-s + 19-s + 3.60·23-s + 7.99·25-s + 7.21·29-s − 9·31-s − 37-s − 10.8·41-s + 8·43-s + 12.9·55-s + 14.4·59-s − 21.6·65-s − 2·67-s + 3.60·71-s − 4·73-s + 7.21·83-s + 25.9·85-s − 10.8·89-s + 3.60·95-s + 8·97-s − 7.21·101-s + 7·103-s − 11·109-s + 7.21·113-s + ⋯ |
| L(s) = 1 | + 1.61·5-s + 1.08·11-s − 1.66·13-s + 1.74·17-s + 0.229·19-s + 0.751·23-s + 1.59·25-s + 1.33·29-s − 1.61·31-s − 0.164·37-s − 1.68·41-s + 1.21·43-s + 1.75·55-s + 1.87·59-s − 2.68·65-s − 0.244·67-s + 0.427·71-s − 0.468·73-s + 0.791·83-s + 2.82·85-s − 1.14·89-s + 0.369·95-s + 0.812·97-s − 0.717·101-s + 0.689·103-s − 1.05·109-s + 0.678·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.054037041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.054037041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.60T + 5T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 - 7.21T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 7.21T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.21T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.265819167829040657970633102951, −7.20327706114244854150645140178, −6.87838502219550164928265239736, −5.89804231652478839396182995802, −5.36930834574889259425878804631, −4.77687034823669286313891869881, −3.56666243931429283557001206609, −2.72649345430817885094586686287, −1.86784613356698102730466432717, −1.00671490868001540260237165256,
1.00671490868001540260237165256, 1.86784613356698102730466432717, 2.72649345430817885094586686287, 3.56666243931429283557001206609, 4.77687034823669286313891869881, 5.36930834574889259425878804631, 5.89804231652478839396182995802, 6.87838502219550164928265239736, 7.20327706114244854150645140178, 8.265819167829040657970633102951
