| L(s) = 1 | − 3·3-s − 11.4·5-s + 9·9-s − 52.5·11-s + 5.48·13-s + 34.3·15-s − 85.3·17-s + 110.·19-s − 209.·23-s + 5.99·25-s − 27·27-s − 132.·29-s + 49.3·31-s + 157.·33-s + 160.·37-s − 16.4·39-s − 138.·41-s − 365.·43-s − 103.·45-s − 131.·47-s + 256.·51-s − 561.·53-s + 601.·55-s − 331.·57-s + 436.·59-s + 291.·61-s − 62.7·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.02·5-s + 0.333·9-s − 1.44·11-s + 0.116·13-s + 0.591·15-s − 1.21·17-s + 1.33·19-s − 1.89·23-s + 0.0479·25-s − 0.192·27-s − 0.850·29-s + 0.285·31-s + 0.832·33-s + 0.712·37-s − 0.0675·39-s − 0.526·41-s − 1.29·43-s − 0.341·45-s − 0.407·47-s + 0.703·51-s − 1.45·53-s + 1.47·55-s − 0.771·57-s + 0.962·59-s + 0.612·61-s − 0.119·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4399328906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4399328906\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 11.4T + 125T^{2} \) |
| 11 | \( 1 + 52.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 5.48T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 209.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 132.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 49.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 160.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 138.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 131.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 561.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 436.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 593.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 775.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 330.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 243.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 332.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 979.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 466.T + 9.12e5T^{2} \) |
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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
−9.569637890404238493236897792017, −8.259155459801771210400630862181, −7.84864201007668684979293588796, −7.00174550936396784581466722126, −5.96876986041445407481915089934, −5.10341602566226007006706319083, −4.25659714257570027835040484075, −3.27829915966571566884209773419, −1.99193214617218706797531556679, −0.33504412875355610270142194740,
0.33504412875355610270142194740, 1.99193214617218706797531556679, 3.27829915966571566884209773419, 4.25659714257570027835040484075, 5.10341602566226007006706319083, 5.96876986041445407481915089934, 7.00174550936396784581466722126, 7.84864201007668684979293588796, 8.259155459801771210400630862181, 9.569637890404238493236897792017
