| L(s) = 1 | − 2-s − 0.00568·3-s + 4-s − 2.34·5-s + 0.00568·6-s − 0.899·7-s − 8-s − 2.99·9-s + 2.34·10-s − 6.58·11-s − 0.00568·12-s + 6.94·13-s + 0.899·14-s + 0.0133·15-s + 16-s − 5.38·17-s + 2.99·18-s + 1.27·19-s − 2.34·20-s + 0.00511·21-s + 6.58·22-s + 1.82·23-s + 0.00568·24-s + 0.498·25-s − 6.94·26-s + 0.0340·27-s − 0.899·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.00328·3-s + 0.5·4-s − 1.04·5-s + 0.00231·6-s − 0.340·7-s − 0.353·8-s − 0.999·9-s + 0.741·10-s − 1.98·11-s − 0.00164·12-s + 1.92·13-s + 0.240·14-s + 0.00344·15-s + 0.250·16-s − 1.30·17-s + 0.707·18-s + 0.292·19-s − 0.524·20-s + 0.00111·21-s + 1.40·22-s + 0.379·23-s + 0.00115·24-s + 0.0997·25-s − 1.36·26-s + 0.00656·27-s − 0.170·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3117531153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3117531153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 0.00568T + 3T^{2} \) |
| 5 | \( 1 + 2.34T + 5T^{2} \) |
| 7 | \( 1 + 0.899T + 7T^{2} \) |
| 11 | \( 1 + 6.58T + 11T^{2} \) |
| 13 | \( 1 - 6.94T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 + 0.353T + 41T^{2} \) |
| 43 | \( 1 + 5.85T + 43T^{2} \) |
| 47 | \( 1 + 8.15T + 47T^{2} \) |
| 53 | \( 1 + 8.20T + 53T^{2} \) |
| 59 | \( 1 - 0.772T + 59T^{2} \) |
| 61 | \( 1 + 4.43T + 61T^{2} \) |
| 67 | \( 1 - 9.34T + 67T^{2} \) |
| 71 | \( 1 - 2.05T + 71T^{2} \) |
| 73 | \( 1 - 0.876T + 73T^{2} \) |
| 79 | \( 1 - 0.108T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.91T + 89T^{2} \) |
| 97 | \( 1 - 6.58T + 97T^{2} \) |
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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
−8.268456576430908198320211577633, −8.085535653178222668908686120202, −7.12880760516268463391114441252, −6.33830500121733398277676823124, −5.59096608950507675787260584604, −4.72820017129257992161199793209, −3.49691317737582116519232203984, −3.07556683051721247554061861919, −1.91817540579639446473858294145, −0.33499670617523964507377773934,
0.33499670617523964507377773934, 1.91817540579639446473858294145, 3.07556683051721247554061861919, 3.49691317737582116519232203984, 4.72820017129257992161199793209, 5.59096608950507675787260584604, 6.33830500121733398277676823124, 7.12880760516268463391114441252, 8.085535653178222668908686120202, 8.268456576430908198320211577633
