| L(s) = 1 | − 3·3-s − 4.63·7-s + 9·9-s + 58.7·11-s + 33.5·13-s + 44.2·17-s − 133.·19-s + 13.9·21-s + 90.0·23-s − 27·27-s + 154.·29-s + 21.0·31-s − 176.·33-s − 38.0·37-s − 100.·39-s + 335.·41-s + 388.·43-s + 267.·47-s − 321.·49-s − 132.·51-s + 445.·53-s + 400.·57-s + 36.6·59-s − 813.·61-s − 41.7·63-s − 41.5·67-s − 270.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.250·7-s + 0.333·9-s + 1.61·11-s + 0.716·13-s + 0.630·17-s − 1.61·19-s + 0.144·21-s + 0.816·23-s − 0.192·27-s + 0.989·29-s + 0.122·31-s − 0.930·33-s − 0.169·37-s − 0.413·39-s + 1.27·41-s + 1.37·43-s + 0.829·47-s − 0.937·49-s − 0.364·51-s + 1.15·53-s + 0.930·57-s + 0.0807·59-s − 1.70·61-s − 0.0834·63-s − 0.0758·67-s − 0.471·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.134722604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.134722604\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.63T + 343T^{2} \) |
| 11 | \( 1 - 58.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 21.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 445.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 + 813.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 41.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 9.99e2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 56.3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 80.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 577.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 193.T + 9.12e5T^{2} \) |
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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
−8.866301949067173654346186691473, −7.79687830830776744604422694177, −6.83643667685005221488103159259, −6.30803186626244717884036063016, −5.69717503394423077813742007974, −4.44131477076480185065715496065, −3.98850780147773582869508625192, −2.85386666396273912998722461978, −1.52786905656358217982186267452, −0.72468688423313199035271002321,
0.72468688423313199035271002321, 1.52786905656358217982186267452, 2.85386666396273912998722461978, 3.98850780147773582869508625192, 4.44131477076480185065715496065, 5.69717503394423077813742007974, 6.30803186626244717884036063016, 6.83643667685005221488103159259, 7.79687830830776744604422694177, 8.866301949067173654346186691473
