| L(s) = 1 | + 0.669·3-s + 4.30·5-s − 2.55·9-s − 0.669·11-s − 13-s + 2.88·15-s − 6.22·17-s − 6.30·19-s − 5.63·23-s + 13.5·25-s − 3.71·27-s − 1.42·29-s − 4.09·31-s − 0.448·33-s + 2.66·37-s − 0.669·39-s − 7.05·41-s − 10.5·43-s − 10.9·45-s − 4.09·47-s − 4.16·51-s + 5.42·53-s − 2.88·55-s − 4.22·57-s + 7.26·59-s − 2.28·61-s − 4.30·65-s + ⋯ |
| L(s) = 1 | + 0.386·3-s + 1.92·5-s − 0.850·9-s − 0.201·11-s − 0.277·13-s + 0.744·15-s − 1.50·17-s − 1.44·19-s − 1.17·23-s + 2.70·25-s − 0.715·27-s − 0.264·29-s − 0.734·31-s − 0.0780·33-s + 0.438·37-s − 0.107·39-s − 1.10·41-s − 1.60·43-s − 1.63·45-s − 0.596·47-s − 0.583·51-s + 0.744·53-s − 0.388·55-s − 0.559·57-s + 0.946·59-s − 0.292·61-s − 0.533·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 0.669T + 3T^{2} \) |
| 5 | \( 1 - 4.30T + 5T^{2} \) |
| 11 | \( 1 + 0.669T + 11T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 5.63T + 23T^{2} \) |
| 29 | \( 1 + 1.42T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 4.09T + 47T^{2} \) |
| 53 | \( 1 - 5.42T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 0.312T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 4.12T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−8.141912050133165440028963799327, −6.85079081429221857449102311188, −6.43343479320609936885899977369, −5.72568094802411297335446223072, −5.11229867272539994584680235799, −4.18835538012165442295849767140, −3.02260281150803862531807752759, −2.12289444668000682002764345962, −1.91781239781914175265985492473, 0,
1.91781239781914175265985492473, 2.12289444668000682002764345962, 3.02260281150803862531807752759, 4.18835538012165442295849767140, 5.11229867272539994584680235799, 5.72568094802411297335446223072, 6.43343479320609936885899977369, 6.85079081429221857449102311188, 8.141912050133165440028963799327
