| L(s) = 1 | − 1.22·5-s + 4.88·7-s − 5.50·11-s − 4.95·13-s − 17-s − 1.70·19-s − 0.614·23-s − 3.50·25-s − 8.17·29-s + 4.88·31-s − 5.98·35-s + 7.27·37-s + 6.50·41-s + 7.68·43-s + 13.1·47-s + 16.9·49-s + 9.45·53-s + 6.73·55-s − 2.56·59-s + 9.72·61-s + 6.05·65-s + 5.98·67-s + 2.70·71-s + 6·73-s − 26.9·77-s − 6.11·79-s + 2.56·83-s + ⋯ |
| L(s) = 1 | − 0.547·5-s + 1.84·7-s − 1.65·11-s − 1.37·13-s − 0.242·17-s − 0.391·19-s − 0.128·23-s − 0.700·25-s − 1.51·29-s + 0.878·31-s − 1.01·35-s + 1.19·37-s + 1.01·41-s + 1.17·43-s + 1.92·47-s + 2.41·49-s + 1.29·53-s + 0.907·55-s − 0.334·59-s + 1.24·61-s + 0.751·65-s + 0.730·67-s + 0.320·71-s + 0.702·73-s − 3.06·77-s − 0.688·79-s + 0.281·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.572107118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572107118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 1.22T + 5T^{2} \) |
| 7 | \( 1 - 4.88T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 + 0.614T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 - 4.88T + 31T^{2} \) |
| 37 | \( 1 - 7.27T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 - 9.45T + 53T^{2} \) |
| 59 | \( 1 + 2.56T + 59T^{2} \) |
| 61 | \( 1 - 9.72T + 61T^{2} \) |
| 67 | \( 1 - 5.98T + 67T^{2} \) |
| 71 | \( 1 - 2.70T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 6.11T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 7.90T + 89T^{2} \) |
| 97 | \( 1 + 5.45T + 97T^{2} \) |
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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
−8.055234079171421332963202751818, −7.56798273267841285009589916266, −7.31042934682303186251387662788, −5.78307920967146221901580676535, −5.31492484525470320720395546522, −4.53335487444906044164148153614, −4.04867316451193077004435617410, −2.45300231244691862207821784579, −2.23970029681849672220143838468, −0.66600150271890057929699067779,
0.66600150271890057929699067779, 2.23970029681849672220143838468, 2.45300231244691862207821784579, 4.04867316451193077004435617410, 4.53335487444906044164148153614, 5.31492484525470320720395546522, 5.78307920967146221901580676535, 7.31042934682303186251387662788, 7.56798273267841285009589916266, 8.055234079171421332963202751818
