| L(s) = 1 | + 3.61·3-s − 5·5-s + 7·7-s − 13.9·9-s − 68.4·11-s + 91.8·13-s − 18.0·15-s + 77.1·17-s − 55.5·19-s + 25.2·21-s − 97.8·23-s + 25·25-s − 147.·27-s + 20.3·29-s − 29.7·31-s − 247.·33-s − 35·35-s + 133.·37-s + 331.·39-s + 99.8·41-s + 298.·43-s + 69.6·45-s + 353.·47-s + 49·49-s + 278.·51-s − 501.·53-s + 342.·55-s + ⋯ |
| L(s) = 1 | + 0.695·3-s − 0.447·5-s + 0.377·7-s − 0.516·9-s − 1.87·11-s + 1.95·13-s − 0.311·15-s + 1.10·17-s − 0.670·19-s + 0.262·21-s − 0.887·23-s + 0.200·25-s − 1.05·27-s + 0.130·29-s − 0.172·31-s − 1.30·33-s − 0.169·35-s + 0.591·37-s + 1.36·39-s + 0.380·41-s + 1.05·43-s + 0.230·45-s + 1.09·47-s + 0.142·49-s + 0.765·51-s − 1.29·53-s + 0.838·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.216336521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.216336521\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 3 | \( 1 - 3.61T + 27T^{2} \) |
| 11 | \( 1 + 68.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 91.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 20.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 29.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 133.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 99.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 298.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 353.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 501.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 638.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 165.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 480.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 28.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 367.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 121.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 65.1T + 9.12e5T^{2} \) |
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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
−8.507667473286162476949494898684, −7.894726291570960918054916341539, −7.64897761849668926936454064235, −6.10202227656284747087177086846, −5.69036941496381106886142960759, −4.57312407080089825837674861717, −3.61215436915000769378072472470, −2.93192825938238716012178908439, −1.95375325753614988450958731553, −0.63276309394581083436649328531,
0.63276309394581083436649328531, 1.95375325753614988450958731553, 2.93192825938238716012178908439, 3.61215436915000769378072472470, 4.57312407080089825837674861717, 5.69036941496381106886142960759, 6.10202227656284747087177086846, 7.64897761849668926936454064235, 7.894726291570960918054916341539, 8.507667473286162476949494898684
