| L(s) = 1 | + 2·2-s + 4·4-s + 19.7·5-s + 8·8-s + 39.5·10-s + 14·11-s + 50.9·13-s + 16·16-s − 1.41·17-s − 1.41·19-s + 79.1·20-s + 28·22-s − 140·23-s + 267·25-s + 101.·26-s + 286·29-s − 93.3·31-s + 32·32-s − 2.82·34-s − 38·37-s − 2.82·38-s + 158.·40-s + 125.·41-s − 34·43-s + 56·44-s − 280·46-s − 523.·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.77·5-s + 0.353·8-s + 1.25·10-s + 0.383·11-s + 1.08·13-s + 0.250·16-s − 0.0201·17-s − 0.0170·19-s + 0.885·20-s + 0.271·22-s − 1.26·23-s + 2.13·25-s + 0.768·26-s + 1.83·29-s − 0.540·31-s + 0.176·32-s − 0.0142·34-s − 0.168·37-s − 0.0120·38-s + 0.626·40-s + 0.479·41-s − 0.120·43-s + 0.191·44-s − 0.897·46-s − 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.135176434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.135176434\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 19.7T + 125T^{2} \) |
| 11 | \( 1 - 14T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 1.41T + 4.91e3T^{2} \) |
| 19 | \( 1 + 1.41T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140T + 1.21e4T^{2} \) |
| 29 | \( 1 - 286T + 2.43e4T^{2} \) |
| 31 | \( 1 + 93.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 34T + 7.95e4T^{2} \) |
| 47 | \( 1 + 523.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 74T + 1.48e5T^{2} \) |
| 59 | \( 1 + 434.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 14.1T + 2.26e5T^{2} \) |
| 67 | \( 1 - 684T + 3.00e5T^{2} \) |
| 71 | \( 1 + 588T + 3.57e5T^{2} \) |
| 73 | \( 1 + 270.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 422.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 618.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−9.898343925728777443595474818229, −9.014929961991690208959117579016, −8.119705103718236217073820959508, −6.68241232975560193979364044125, −6.21924781091340677971233010124, −5.49922180824754197246436543132, −4.48845378918428818031632182992, −3.27655088289964467490530643500, −2.15437985690133539072925133592, −1.26920257380822685409214092550,
1.26920257380822685409214092550, 2.15437985690133539072925133592, 3.27655088289964467490530643500, 4.48845378918428818031632182992, 5.49922180824754197246436543132, 6.21924781091340677971233010124, 6.68241232975560193979364044125, 8.119705103718236217073820959508, 9.014929961991690208959117579016, 9.898343925728777443595474818229
