| L(s) = 1 | + 5.26·2-s − 3·3-s + 19.6·4-s − 5·5-s − 15.7·6-s + 10.3·7-s + 61.4·8-s + 9·9-s − 26.3·10-s − 59.0·12-s − 63.9·13-s + 54.3·14-s + 15·15-s + 165.·16-s − 17.1·17-s + 47.3·18-s − 90.2·19-s − 98.4·20-s − 30.9·21-s − 212.·23-s − 184.·24-s + 25·25-s − 336.·26-s − 27·27-s + 203.·28-s − 57.5·29-s + 78.9·30-s + ⋯ |
| L(s) = 1 | + 1.86·2-s − 0.577·3-s + 2.46·4-s − 0.447·5-s − 1.07·6-s + 0.557·7-s + 2.71·8-s + 0.333·9-s − 0.831·10-s − 1.42·12-s − 1.36·13-s + 1.03·14-s + 0.258·15-s + 2.59·16-s − 0.244·17-s + 0.620·18-s − 1.08·19-s − 1.10·20-s − 0.321·21-s − 1.92·23-s − 1.56·24-s + 0.200·25-s − 2.53·26-s − 0.192·27-s + 1.37·28-s − 0.368·29-s + 0.480·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 5.26T + 8T^{2} \) |
| 7 | \( 1 - 10.3T + 343T^{2} \) |
| 13 | \( 1 + 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 17.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 90.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 212.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 141.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 225.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 347.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 404.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 259.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 853.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 266.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 92.4T + 3.57e5T^{2} \) |
| 73 | \( 1 - 242.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 706.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 440.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 197.T + 9.12e5T^{2} \) |
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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
−8.072754521640923200279475969157, −7.40711684547733782273394111388, −6.67979398888853957502572180066, −5.81147372032933754121635651047, −5.19038654321030509811003287300, −4.31009615260131408553926853374, −3.93830525393156702777458907153, −2.55309200048611741905301304638, −1.82971697880640386232104031218, 0,
1.82971697880640386232104031218, 2.55309200048611741905301304638, 3.93830525393156702777458907153, 4.31009615260131408553926853374, 5.19038654321030509811003287300, 5.81147372032933754121635651047, 6.67979398888853957502572180066, 7.40711684547733782273394111388, 8.072754521640923200279475969157
