| L(s) = 1 | + 3·5-s + 3.30·7-s + 6.30·13-s + 1.30·17-s + 2·19-s − 1.30·23-s + 4·25-s − 0.394·29-s − 0.605·31-s + 9.90·35-s + 7.60·37-s − 8.21·41-s + 2.39·43-s + 5.60·47-s + 3.90·49-s − 2.60·53-s + 13.8·59-s − 14.8·61-s + 18.9·65-s − 6.21·67-s + 6.90·71-s − 13.9·73-s + 13.6·79-s − 5.60·83-s + 3.90·85-s − 7.81·89-s + 20.8·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.24·7-s + 1.74·13-s + 0.315·17-s + 0.458·19-s − 0.271·23-s + 0.800·25-s − 0.0732·29-s − 0.108·31-s + 1.67·35-s + 1.25·37-s − 1.28·41-s + 0.365·43-s + 0.817·47-s + 0.558·49-s − 0.357·53-s + 1.79·59-s − 1.89·61-s + 2.34·65-s − 0.758·67-s + 0.819·71-s − 1.62·73-s + 1.53·79-s − 0.615·83-s + 0.423·85-s − 0.828·89-s + 2.18·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.672965610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.672965610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 - 1.30T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 + 0.394T + 29T^{2} \) |
| 31 | \( 1 + 0.605T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 + 8.21T + 41T^{2} \) |
| 43 | \( 1 - 2.39T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 + 2.60T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 6.21T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.187282186472679021982844735090, −7.44346826825739842210961240083, −6.47512391492057174641361757289, −5.85936947209048348045760151659, −5.39367056971665751879451724818, −4.53044945036982782249640413760, −3.67165279116746009034227060832, −2.62723449833689006743733027752, −1.66328674420939585085062941828, −1.17500671843368084589159178289,
1.17500671843368084589159178289, 1.66328674420939585085062941828, 2.62723449833689006743733027752, 3.67165279116746009034227060832, 4.53044945036982782249640413760, 5.39367056971665751879451724818, 5.85936947209048348045760151659, 6.47512391492057174641361757289, 7.44346826825739842210961240083, 8.187282186472679021982844735090
