| L(s) = 1 | + 0.929·2-s + 0.505·3-s − 7.13·4-s + 10.4·5-s + 0.469·6-s + 30.2·7-s − 14.0·8-s − 26.7·9-s + 9.75·10-s − 36.7·11-s − 3.60·12-s − 65.2·13-s + 28.1·14-s + 5.30·15-s + 44.0·16-s + 36.6·17-s − 24.8·18-s − 42.1·19-s − 74.8·20-s + 15.2·21-s − 34.1·22-s − 7.10·24-s − 14.8·25-s − 60.6·26-s − 27.1·27-s − 215.·28-s + 57.5·29-s + ⋯ |
| L(s) = 1 | + 0.328·2-s + 0.0972·3-s − 0.891·4-s + 0.938·5-s + 0.0319·6-s + 1.63·7-s − 0.621·8-s − 0.990·9-s + 0.308·10-s − 1.00·11-s − 0.0867·12-s − 1.39·13-s + 0.536·14-s + 0.0912·15-s + 0.687·16-s + 0.523·17-s − 0.325·18-s − 0.509·19-s − 0.837·20-s + 0.158·21-s − 0.331·22-s − 0.0604·24-s − 0.118·25-s − 0.457·26-s − 0.193·27-s − 1.45·28-s + 0.368·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 0.929T + 8T^{2} \) |
| 3 | \( 1 - 0.505T + 27T^{2} \) |
| 5 | \( 1 - 10.4T + 125T^{2} \) |
| 7 | \( 1 - 30.2T + 343T^{2} \) |
| 11 | \( 1 + 36.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 65.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.1T + 6.85e3T^{2} \) |
| 29 | \( 1 - 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 58.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 288.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 350.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 186.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 145.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 415.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 949.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 417.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 82.7T + 4.93e5T^{2} \) |
| 83 | \( 1 + 71.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 737.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−10.03064362032176711796576294153, −9.025425833754205609237078079662, −8.279392201063691575125504094595, −7.51741895647392930313195668221, −5.81617609119119387717840994688, −5.22439506569362079305675271061, −4.58934332285957841917359970844, −2.93156159731495830095698666828, −1.83370770159871589885723971770, 0,
1.83370770159871589885723971770, 2.93156159731495830095698666828, 4.58934332285957841917359970844, 5.22439506569362079305675271061, 5.81617609119119387717840994688, 7.51741895647392930313195668221, 8.279392201063691575125504094595, 9.025425833754205609237078079662, 10.03064362032176711796576294153
