| 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
| \(L(2)\) |
\(\approx\) |
\(2.841192368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.841192368\) |
| \(L(\frac{5}{2})\) |
|
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
−8.382477846900231906990823414425, −7.938097382420788839905363338584, −7.35561470024418069758944376003, −6.24754433370863703241988890159, −5.38090099915986629269871367280, −4.73060880151125697312629371384, −3.51596839473190566480402681414, −2.91514146890939884958771245939, −1.85479608364255522408034293963, −0.73148764992600026174119315070,
0.73148764992600026174119315070, 1.85479608364255522408034293963, 2.91514146890939884958771245939, 3.51596839473190566480402681414, 4.73060880151125697312629371384, 5.38090099915986629269871367280, 6.24754433370863703241988890159, 7.35561470024418069758944376003, 7.938097382420788839905363338584, 8.382477846900231906990823414425
