| L(s) = 1 | + 2.49·2-s + 4.22·4-s + 3.40·5-s − 2.82·7-s + 5.55·8-s + 8.48·10-s − 11-s + 4.13·13-s − 7.05·14-s + 5.41·16-s − 6.64·17-s + 0.858·19-s + 14.3·20-s − 2.49·22-s − 4.06·23-s + 6.56·25-s + 10.3·26-s − 11.9·28-s + 10.2·29-s − 3.76·31-s + 2.40·32-s − 16.5·34-s − 9.61·35-s − 8.11·37-s + 2.14·38-s + 18.9·40-s + 1.50·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 2.11·4-s + 1.52·5-s − 1.06·7-s + 1.96·8-s + 2.68·10-s − 0.301·11-s + 1.14·13-s − 1.88·14-s + 1.35·16-s − 1.61·17-s + 0.196·19-s + 3.21·20-s − 0.532·22-s − 0.846·23-s + 1.31·25-s + 2.02·26-s − 2.25·28-s + 1.90·29-s − 0.675·31-s + 0.424·32-s − 2.84·34-s − 1.62·35-s − 1.33·37-s + 0.347·38-s + 2.98·40-s + 0.234·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.917632262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.917632262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.49T + 2T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 13 | \( 1 - 4.13T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 - 0.858T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 1.50T + 41T^{2} \) |
| 43 | \( 1 - 1.48T + 43T^{2} \) |
| 47 | \( 1 + 8.51T + 47T^{2} \) |
| 53 | \( 1 + 0.252T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.334T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 9.77T + 71T^{2} \) |
| 73 | \( 1 - 7.29T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 8.31T + 83T^{2} \) |
| 89 | \( 1 - 1.65T + 89T^{2} \) |
| 97 | \( 1 + 9.18T + 97T^{2} \) |
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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
−10.36024401741891287867615848070, −9.465771163718354460155609154212, −8.493380244913484947655253171968, −6.77153782557984402385660850857, −6.45595797416088265757091165604, −5.76864140714726330098444218600, −4.89962377984455462450503877146, −3.79151984697612866138947865656, −2.81164556975075910885690533986, −1.90386177366371464256103671398,
1.90386177366371464256103671398, 2.81164556975075910885690533986, 3.79151984697612866138947865656, 4.89962377984455462450503877146, 5.76864140714726330098444218600, 6.45595797416088265757091165604, 6.77153782557984402385660850857, 8.493380244913484947655253171968, 9.465771163718354460155609154212, 10.36024401741891287867615848070
