| L(s) = 1 | − 2.52·2-s + 3-s + 4.36·4-s − 5-s − 2.52·6-s + 7-s − 5.96·8-s + 9-s + 2.52·10-s + 11-s + 4.36·12-s + 5.44·13-s − 2.52·14-s − 15-s + 6.31·16-s + 1.41·17-s − 2.52·18-s − 1.36·19-s − 4.36·20-s + 21-s − 2.52·22-s − 8.40·23-s − 5.96·24-s + 25-s − 13.7·26-s + 27-s + 4.36·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.18·4-s − 0.447·5-s − 1.02·6-s + 0.377·7-s − 2.10·8-s + 0.333·9-s + 0.797·10-s + 0.301·11-s + 1.25·12-s + 1.50·13-s − 0.674·14-s − 0.258·15-s + 1.57·16-s + 0.342·17-s − 0.594·18-s − 0.312·19-s − 0.975·20-s + 0.218·21-s − 0.537·22-s − 1.75·23-s − 1.21·24-s + 0.200·25-s − 2.69·26-s + 0.192·27-s + 0.824·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9148611933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9148611933\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 13 | \( 1 - 5.44T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 + 8.40T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 8.21T + 47T^{2} \) |
| 53 | \( 1 + 5.31T + 53T^{2} \) |
| 59 | \( 1 - 7.07T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 - 4.69T + 71T^{2} \) |
| 73 | \( 1 - 7.31T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 + 9.51T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.782163170988500597650362520968, −8.737454662288425632983810825980, −8.249571398454788893330217938051, −7.87608604951895777392310152312, −6.72344415158763329380402199399, −6.09690868727350506185242526080, −4.37520801385368843504160525730, −3.26856566858543446774365480886, −2.01525763931295709078079235613, −0.958313057404705504034689164102,
0.958313057404705504034689164102, 2.01525763931295709078079235613, 3.26856566858543446774365480886, 4.37520801385368843504160525730, 6.09690868727350506185242526080, 6.72344415158763329380402199399, 7.87608604951895777392310152312, 8.249571398454788893330217938051, 8.737454662288425632983810825980, 9.782163170988500597650362520968
