| L(s) = 1 | − 3.27·5-s − 3.27·11-s + 6.27·13-s + 4·17-s − 6.27·19-s − 4·23-s + 5.72·25-s − 5.27·29-s − 31-s − 2.27·37-s + 4.54·41-s + 0.274·43-s + 6·47-s − 9.27·53-s + 10.7·55-s + 1.27·59-s + 10·61-s − 20.5·65-s − 0.274·67-s − 2·71-s − 4.27·73-s + 11.5·79-s − 7.27·83-s − 13.0·85-s + 10.5·89-s + 20.5·95-s + 8.72·97-s + ⋯ |
| L(s) = 1 | − 1.46·5-s − 0.987·11-s + 1.74·13-s + 0.970·17-s − 1.43·19-s − 0.834·23-s + 1.14·25-s − 0.979·29-s − 0.179·31-s − 0.373·37-s + 0.710·41-s + 0.0419·43-s + 0.875·47-s − 1.27·53-s + 1.44·55-s + 0.165·59-s + 1.28·61-s − 2.54·65-s − 0.0335·67-s − 0.237·71-s − 0.500·73-s + 1.29·79-s − 0.798·83-s − 1.42·85-s + 1.11·89-s + 2.10·95-s + 0.885·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059603996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059603996\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 13 | \( 1 - 6.27T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 6.27T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 5.27T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 0.274T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 9.27T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 0.274T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 4.27T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 7.27T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 8.72T + 97T^{2} \) |
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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
−8.386331879506744594668725255358, −7.907935139283197686345424188363, −7.32971537217748960796750790196, −6.27103055964608145630893214925, −5.65168243120732095731892533736, −4.56428999046448773875057831504, −3.81891220525338338586895157424, −3.31714133186533196958616598180, −2.00740241212373459161417113774, −0.59489909251531065453496301514,
0.59489909251531065453496301514, 2.00740241212373459161417113774, 3.31714133186533196958616598180, 3.81891220525338338586895157424, 4.56428999046448773875057831504, 5.65168243120732095731892533736, 6.27103055964608145630893214925, 7.32971537217748960796750790196, 7.907935139283197686345424188363, 8.386331879506744594668725255358
