| L(s) = 1 | + 2.13·2-s + 3-s + 2.55·4-s + 2.13·6-s + 4.82·7-s + 1.17·8-s + 9-s + 2.55·12-s + 4.26·13-s + 10.2·14-s − 2.59·16-s + 4.64·17-s + 2.13·18-s + 6.37·19-s + 4.82·21-s + 5.14·23-s + 1.17·24-s + 9.10·26-s + 27-s + 12.3·28-s − 4.26·29-s − 6.39·31-s − 7.88·32-s + 9.90·34-s + 2.55·36-s − 6.14·37-s + 13.5·38-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 0.577·3-s + 1.27·4-s + 0.871·6-s + 1.82·7-s + 0.416·8-s + 0.333·9-s + 0.736·12-s + 1.18·13-s + 2.74·14-s − 0.647·16-s + 1.12·17-s + 0.502·18-s + 1.46·19-s + 1.05·21-s + 1.07·23-s + 0.240·24-s + 1.78·26-s + 0.192·27-s + 2.32·28-s − 0.792·29-s − 1.14·31-s − 1.39·32-s + 1.69·34-s + 0.425·36-s − 1.00·37-s + 2.20·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.832185649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.832185649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 + 6.14T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 1.39T + 89T^{2} \) |
| 97 | \( 1 + 3.93T + 97T^{2} \) |
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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
−7.49982996338725734662458652946, −7.17559679805485548703712664863, −6.04645949676709645948985159410, −5.34095931314852785961683890871, −5.06632552118570126330331962200, −4.23273172191972809142206028599, −3.45493483471906436623516822933, −3.06502724295358663676345157948, −1.79402911410993912218170999725, −1.32095636579422879766022897776,
1.32095636579422879766022897776, 1.79402911410993912218170999725, 3.06502724295358663676345157948, 3.45493483471906436623516822933, 4.23273172191972809142206028599, 5.06632552118570126330331962200, 5.34095931314852785961683890871, 6.04645949676709645948985159410, 7.17559679805485548703712664863, 7.49982996338725734662458652946
