| L(s) = 1 | + 19.8·2-s + 27·3-s + 267.·4-s + 443.·5-s + 536.·6-s + 343·7-s + 2.76e3·8-s + 729·9-s + 8.80e3·10-s − 3.72e3·11-s + 7.20e3·12-s + 472.·13-s + 6.81e3·14-s + 1.19e4·15-s + 2.07e4·16-s − 1.65e4·17-s + 1.44e4·18-s + 9.33e3·19-s + 1.18e5·20-s + 9.26e3·21-s − 7.41e4·22-s − 1.21e4·23-s + 7.46e4·24-s + 1.18e5·25-s + 9.39e3·26-s + 1.96e4·27-s + 9.15e4·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.08·4-s + 1.58·5-s + 1.01·6-s + 0.377·7-s + 1.90·8-s + 0.333·9-s + 2.78·10-s − 0.844·11-s + 1.20·12-s + 0.0596·13-s + 0.663·14-s + 0.915·15-s + 1.26·16-s − 0.816·17-s + 0.585·18-s + 0.312·19-s + 3.30·20-s + 0.218·21-s − 1.48·22-s − 0.208·23-s + 1.10·24-s + 1.51·25-s + 0.104·26-s + 0.192·27-s + 0.788·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(12.89517344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.89517344\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 7 | \( 1 - 343T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 19.8T + 128T^{2} \) |
| 5 | \( 1 - 443.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.72e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 472.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.65e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.33e3T + 8.93e8T^{2} \) |
| 29 | \( 1 - 1.97e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.07e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.35e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 585.T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.69e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.61e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.97e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.51e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.58e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 6.16e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.22e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 9.81e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.54e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.09e7T + 8.07e13T^{2} \) |
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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
−10.09069802495699013166875538607, −8.975385169607335276977484194191, −7.80555875431260844620390223049, −6.59830269568882515284439709674, −5.98217279913867916893262238511, −5.02154613512952462970023972173, −4.34642585286650104559019300712, −2.78473458528732187444324893654, −2.48062752987316689286054654424, −1.33937158515293688605828294685,
1.33937158515293688605828294685, 2.48062752987316689286054654424, 2.78473458528732187444324893654, 4.34642585286650104559019300712, 5.02154613512952462970023972173, 5.98217279913867916893262238511, 6.59830269568882515284439709674, 7.80555875431260844620390223049, 8.975385169607335276977484194191, 10.09069802495699013166875538607
