| L(s) = 1 | + 2.34·3-s + 5-s + 1.19·7-s + 2.48·9-s − 4.97·11-s + 6.63·13-s + 2.34·15-s + 1.48·17-s − 19-s + 2.80·21-s − 0.510·23-s + 25-s − 1.19·27-s + 7.88·29-s − 2.97·31-s − 11.6·33-s + 1.19·35-s + 7.14·37-s + 15.5·39-s + 1.66·41-s + 6.39·43-s + 2.48·45-s − 9.95·47-s − 5.56·49-s + 3.48·51-s + 11.4·53-s − 4.97·55-s + ⋯ |
| L(s) = 1 | + 1.35·3-s + 0.447·5-s + 0.452·7-s + 0.829·9-s − 1.50·11-s + 1.84·13-s + 0.604·15-s + 0.361·17-s − 0.229·19-s + 0.611·21-s − 0.106·23-s + 0.200·25-s − 0.230·27-s + 1.46·29-s − 0.534·31-s − 2.03·33-s + 0.202·35-s + 1.17·37-s + 2.48·39-s + 0.259·41-s + 0.974·43-s + 0.371·45-s − 1.45·47-s − 0.795·49-s + 0.488·51-s + 1.56·53-s − 0.671·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.000784067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.000784067\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.34T + 3T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 6.63T + 13T^{2} \) |
| 17 | \( 1 - 1.48T + 17T^{2} \) |
| 23 | \( 1 + 0.510T + 23T^{2} \) |
| 29 | \( 1 - 7.88T + 29T^{2} \) |
| 31 | \( 1 + 2.97T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 - 1.66T + 41T^{2} \) |
| 43 | \( 1 - 6.39T + 43T^{2} \) |
| 47 | \( 1 + 9.95T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 8.68T + 83T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−8.209789567657295742328377737456, −7.72409777978309122486627057275, −6.70888284435640576972737198364, −5.90921971615937092889217270074, −5.22629499736552280282287453021, −4.27157168323402117886246749196, −3.46918107391822402028254002480, −2.73972352232129884699292131021, −2.07320124755585031596456274824, −1.02798591366807118343159379613,
1.02798591366807118343159379613, 2.07320124755585031596456274824, 2.73972352232129884699292131021, 3.46918107391822402028254002480, 4.27157168323402117886246749196, 5.22629499736552280282287453021, 5.90921971615937092889217270074, 6.70888284435640576972737198364, 7.72409777978309122486627057275, 8.209789567657295742328377737456
