| L(s) = 1 | + 0.777·2-s + 3-s − 1.39·4-s + 1.77·5-s + 0.777·6-s + 0.717·7-s − 2.64·8-s + 9-s + 1.38·10-s + 1.42·11-s − 1.39·12-s + 6.60·13-s + 0.558·14-s + 1.77·15-s + 0.737·16-s − 17-s + 0.777·18-s − 1.06·19-s − 2.48·20-s + 0.717·21-s + 1.11·22-s − 2.45·23-s − 2.64·24-s − 1.84·25-s + 5.13·26-s + 27-s − 1.00·28-s + ⋯ |
| L(s) = 1 | + 0.549·2-s + 0.577·3-s − 0.697·4-s + 0.794·5-s + 0.317·6-s + 0.271·7-s − 0.933·8-s + 0.333·9-s + 0.437·10-s + 0.430·11-s − 0.402·12-s + 1.83·13-s + 0.149·14-s + 0.458·15-s + 0.184·16-s − 0.242·17-s + 0.183·18-s − 0.245·19-s − 0.554·20-s + 0.156·21-s + 0.236·22-s − 0.511·23-s − 0.538·24-s − 0.368·25-s + 1.00·26-s + 0.192·27-s − 0.189·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.429885676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.429885676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 0.777T + 2T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 0.717T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 - 6.60T + 13T^{2} \) |
| 19 | \( 1 + 1.06T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 - 4.79T + 29T^{2} \) |
| 31 | \( 1 - 5.49T + 31T^{2} \) |
| 37 | \( 1 + 2.95T + 37T^{2} \) |
| 41 | \( 1 - 4.17T + 41T^{2} \) |
| 43 | \( 1 + 3.22T + 43T^{2} \) |
| 47 | \( 1 - 0.511T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 2.89T + 59T^{2} \) |
| 61 | \( 1 - 9.69T + 61T^{2} \) |
| 67 | \( 1 - 7.90T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 83 | \( 1 + 1.13T + 83T^{2} \) |
| 89 | \( 1 + 0.0228T + 89T^{2} \) |
| 97 | \( 1 - 9.55T + 97T^{2} \) |
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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
−8.461156558788540437166567920529, −8.055687545917170328554256785209, −6.66273564290845934779952392619, −6.18933282078738732925568652405, −5.47107249743528705104386873830, −4.53171929672133964035527172288, −3.88670267911841918872487590927, −3.15659519646862898715431965226, −2.03991663616397950932198463237, −1.02289070358790856680955344924,
1.02289070358790856680955344924, 2.03991663616397950932198463237, 3.15659519646862898715431965226, 3.88670267911841918872487590927, 4.53171929672133964035527172288, 5.47107249743528705104386873830, 6.18933282078738732925568652405, 6.66273564290845934779952392619, 8.055687545917170328554256785209, 8.461156558788540437166567920529
