| L(s) = 1 | + 8.86·3-s + 18.8·5-s − 20.8·7-s + 51.5·9-s + 13.7·11-s + 13·13-s + 167.·15-s + 6.86·17-s − 52.6·19-s − 184.·21-s − 188.·23-s + 230.·25-s + 218.·27-s − 206.·29-s + 290.·31-s + 121.·33-s − 393.·35-s − 220.·37-s + 115.·39-s + 149.·41-s + 183.·43-s + 973.·45-s − 146.·47-s + 92.3·49-s + 60.8·51-s + 409.·53-s + 259.·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s + 1.68·5-s − 1.12·7-s + 1.91·9-s + 0.376·11-s + 0.277·13-s + 2.87·15-s + 0.0979·17-s − 0.635·19-s − 1.92·21-s − 1.71·23-s + 1.84·25-s + 1.55·27-s − 1.32·29-s + 1.68·31-s + 0.642·33-s − 1.90·35-s − 0.978·37-s + 0.473·39-s + 0.570·41-s + 0.649·43-s + 3.22·45-s − 0.454·47-s + 0.269·49-s + 0.167·51-s + 1.06·53-s + 0.635·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.620584078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.620584078\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 - 8.86T + 27T^{2} \) |
| 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 + 20.8T + 343T^{2} \) |
| 11 | \( 1 - 13.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 6.86T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 290.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 451.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 247.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 472.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 478.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 411.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−12.39081145021428496483774024158, −10.39926316535134649787537948140, −9.692411141679865214281199365864, −9.189329229931004300173001143909, −8.178170319683892287518863358443, −6.76475556380773691217029819378, −5.89083272783891321497668812139, −3.97323479192961917854624765732, −2.75940972815044702338944391627, −1.79558986910770657751440976934,
1.79558986910770657751440976934, 2.75940972815044702338944391627, 3.97323479192961917854624765732, 5.89083272783891321497668812139, 6.76475556380773691217029819378, 8.178170319683892287518863358443, 9.189329229931004300173001143909, 9.692411141679865214281199365864, 10.39926316535134649787537948140, 12.39081145021428496483774024158
