| L(s) = 1 | − 21.0·3-s − 22.6·5-s − 84.1·7-s + 200.·9-s + 121.·11-s − 253.·13-s + 476.·15-s − 1.01e3·17-s − 1.21e3·19-s + 1.77e3·21-s − 98.9·23-s − 2.61e3·25-s + 889.·27-s + 1.04e3·29-s − 6.65e3·31-s − 2.55e3·33-s + 1.90e3·35-s − 1.25e4·37-s + 5.34e3·39-s − 2.48e3·41-s − 2.29e4·43-s − 4.54e3·45-s + 2.94e3·47-s − 9.72e3·49-s + 2.14e4·51-s − 2.25e4·53-s − 2.74e3·55-s + ⋯ |
| L(s) = 1 | − 1.35·3-s − 0.404·5-s − 0.649·7-s + 0.826·9-s + 0.302·11-s − 0.416·13-s + 0.546·15-s − 0.853·17-s − 0.770·19-s + 0.877·21-s − 0.0390·23-s − 0.836·25-s + 0.234·27-s + 0.230·29-s − 1.24·31-s − 0.408·33-s + 0.262·35-s − 1.50·37-s + 0.562·39-s − 0.230·41-s − 1.88·43-s − 0.334·45-s + 0.194·47-s − 0.578·49-s + 1.15·51-s − 1.10·53-s − 0.122·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1449453210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1449453210\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 67 | \( 1 - 4.48e3T \) |
| good | 3 | \( 1 + 21.0T + 243T^{2} \) |
| 5 | \( 1 + 22.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 84.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 121.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 253.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.21e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 98.9T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.94e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.25e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.85e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.16e4T + 8.44e8T^{2} \) |
| 71 | \( 1 - 1.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.71e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.48e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.81e4T + 8.58e9T^{2} \) |
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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
−10.24892057041763804526451207865, −9.294110918201380670174770668567, −8.237883992710639607472884851702, −6.95366791866093369353013274789, −6.45655603471040666785221585991, −5.44397823473100435735952063686, −4.52912486250353975608290151515, −3.42977043581238572115185511667, −1.82481567065962575135479338254, −0.19533606754255617114378233578,
0.19533606754255617114378233578, 1.82481567065962575135479338254, 3.42977043581238572115185511667, 4.52912486250353975608290151515, 5.44397823473100435735952063686, 6.45655603471040666785221585991, 6.95366791866093369353013274789, 8.237883992710639607472884851702, 9.294110918201380670174770668567, 10.24892057041763804526451207865
