| L(s) = 1 | + 1.14·2-s + 3·3-s − 6.68·4-s + 8.47·5-s + 3.43·6-s + 26.2·7-s − 16.8·8-s + 9·9-s + 9.71·10-s + 32.0·11-s − 20.0·12-s + 44.6·13-s + 30.1·14-s + 25.4·15-s + 34.1·16-s − 79.5·17-s + 10.3·18-s − 144.·19-s − 56.6·20-s + 78.7·21-s + 36.7·22-s + 134.·23-s − 50.5·24-s − 53.2·25-s + 51.1·26-s + 27·27-s − 175.·28-s + ⋯ |
| L(s) = 1 | + 0.405·2-s + 0.577·3-s − 0.835·4-s + 0.757·5-s + 0.234·6-s + 1.41·7-s − 0.744·8-s + 0.333·9-s + 0.307·10-s + 0.879·11-s − 0.482·12-s + 0.951·13-s + 0.574·14-s + 0.437·15-s + 0.533·16-s − 1.13·17-s + 0.135·18-s − 1.74·19-s − 0.633·20-s + 0.818·21-s + 0.356·22-s + 1.21·23-s − 0.429·24-s − 0.425·25-s + 0.385·26-s + 0.192·27-s − 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.281658256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.281658256\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 1.14T + 8T^{2} \) |
| 5 | \( 1 - 8.47T + 125T^{2} \) |
| 7 | \( 1 - 26.2T + 343T^{2} \) |
| 11 | \( 1 - 32.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 144.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 51.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 364.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 254.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 677.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 750.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 614.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 645.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 551.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 440.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 191.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 770.T + 9.12e5T^{2} \) |
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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
−13.67594197771700433653463042279, −13.09635181686352952110950360186, −11.60787139856947581859656733666, −10.39240006207206840014105309149, −8.819827150026105277437141902844, −8.578966190404356334684618972258, −6.57549512121996843107055536809, −5.06431011714212476442800554649, −3.93448197947135051892316652663, −1.76908492588492472794925091010,
1.76908492588492472794925091010, 3.93448197947135051892316652663, 5.06431011714212476442800554649, 6.57549512121996843107055536809, 8.578966190404356334684618972258, 8.819827150026105277437141902844, 10.39240006207206840014105309149, 11.60787139856947581859656733666, 13.09635181686352952110950360186, 13.67594197771700433653463042279
