| L(s) = 1 | − 5.80·5-s − 26.1·7-s − 37.2·11-s − 30.3·13-s − 48.4·17-s + 88.1·19-s + 84.5·23-s − 91.3·25-s + 176.·29-s + 155.·31-s + 151.·35-s − 258.·37-s + 220.·41-s − 47.7·43-s + 129.·47-s + 339.·49-s + 577.·53-s + 216.·55-s − 243.·59-s − 687.·61-s + 176.·65-s + 378.·67-s + 332.·71-s + 1.07e3·73-s + 972.·77-s − 1.26e3·79-s + 1.22e3·83-s + ⋯ |
| L(s) = 1 | − 0.519·5-s − 1.41·7-s − 1.02·11-s − 0.647·13-s − 0.691·17-s + 1.06·19-s + 0.766·23-s − 0.730·25-s + 1.13·29-s + 0.900·31-s + 0.732·35-s − 1.14·37-s + 0.839·41-s − 0.169·43-s + 0.401·47-s + 0.989·49-s + 1.49·53-s + 0.529·55-s − 0.537·59-s − 1.44·61-s + 0.335·65-s + 0.689·67-s + 0.556·71-s + 1.71·73-s + 1.43·77-s − 1.79·79-s + 1.62·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9234591842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9234591842\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.80T + 125T^{2} \) |
| 7 | \( 1 + 26.1T + 343T^{2} \) |
| 11 | \( 1 + 37.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 84.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 176.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 47.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 129.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 577.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 243.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 687.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 378.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.22e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−10.07414869236753903103588896711, −9.434242514882063977919623898431, −8.397631048179690295276674704600, −7.41846584582862698511518083302, −6.72195967636200932754369872587, −5.62586089080563453109431266850, −4.59138808503380607516656098001, −3.35099157182812488973781923918, −2.55718196346138018144349260813, −0.53433121696422723629177386693,
0.53433121696422723629177386693, 2.55718196346138018144349260813, 3.35099157182812488973781923918, 4.59138808503380607516656098001, 5.62586089080563453109431266850, 6.72195967636200932754369872587, 7.41846584582862698511518083302, 8.397631048179690295276674704600, 9.434242514882063977919623898431, 10.07414869236753903103588896711
