L(s) = 1 | − 0.602·3-s + 3.10·7-s − 2.63·9-s + 6.35·11-s + 1.18·13-s − 8.12·17-s + 3.07·19-s − 1.87·21-s − 4.85·23-s + 3.39·27-s + 1.77·29-s − 6.90·31-s − 3.82·33-s + 6.84·37-s − 0.715·39-s − 5.93·41-s − 4.11·43-s − 1.38·47-s + 2.65·49-s + 4.89·51-s − 10.8·53-s − 1.85·57-s − 10.3·59-s + 8.21·61-s − 8.19·63-s + 0.367·67-s + 2.92·69-s + ⋯ |
L(s) = 1 | − 0.347·3-s + 1.17·7-s − 0.879·9-s + 1.91·11-s + 0.329·13-s − 1.97·17-s + 0.706·19-s − 0.408·21-s − 1.01·23-s + 0.653·27-s + 0.329·29-s − 1.24·31-s − 0.666·33-s + 1.12·37-s − 0.114·39-s − 0.926·41-s − 0.626·43-s − 0.201·47-s + 0.378·49-s + 0.685·51-s − 1.49·53-s − 0.245·57-s − 1.34·59-s + 1.05·61-s − 1.03·63-s + 0.0449·67-s + 0.352·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.602T + 3T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 6.35T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 + 8.12T + 17T^{2} \) |
| 19 | \( 1 - 3.07T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 6.90T + 31T^{2} \) |
| 37 | \( 1 - 6.84T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 + 1.38T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 8.21T + 61T^{2} \) |
| 67 | \( 1 - 0.367T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 + 5.05T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 3.71T + 83T^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 + 4.97T + 97T^{2} \) |
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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
−7.22385319688266099393545117044, −6.50139749726416113723075927782, −6.09033577278858618796203328331, −5.24338053023999988005293450768, −4.48697349406640405753153045976, −4.00941910776998967335661511569, −3.03714727809173787816583389908, −1.91931303874710532698478238552, −1.36251584957084959594039619554, 0,
1.36251584957084959594039619554, 1.91931303874710532698478238552, 3.03714727809173787816583389908, 4.00941910776998967335661511569, 4.48697349406640405753153045976, 5.24338053023999988005293450768, 6.09033577278858618796203328331, 6.50139749726416113723075927782, 7.22385319688266099393545117044