| L(s) = 1 | + 3·3-s − 10.3·5-s + 3.46·7-s + 9·9-s + 55.4·13-s − 31.1·15-s − 90·17-s − 116·19-s + 10.3·21-s + 103.·23-s − 17·25-s + 27·27-s + 259.·29-s − 301.·31-s − 36·35-s − 34.6·37-s + 166.·39-s + 54·41-s − 20·43-s − 93.5·45-s + 394.·47-s − 331·49-s − 270·51-s − 488.·53-s − 348·57-s − 324·59-s − 575.·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.929·5-s + 0.187·7-s + 0.333·9-s + 1.18·13-s − 0.536·15-s − 1.28·17-s − 1.40·19-s + 0.107·21-s + 0.942·23-s − 0.136·25-s + 0.192·27-s + 1.66·29-s − 1.74·31-s − 0.173·35-s − 0.153·37-s + 0.682·39-s + 0.205·41-s − 0.0709·43-s − 0.309·45-s + 1.22·47-s − 0.965·49-s − 0.741·51-s − 1.26·53-s − 0.808·57-s − 0.714·59-s − 1.20·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| good | 5 | \( 1 + 10.3T + 125T^{2} \) |
| 7 | \( 1 - 3.46T + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90T + 4.91e3T^{2} \) |
| 19 | \( 1 + 116T + 6.85e3T^{2} \) |
| 23 | \( 1 - 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 259.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 34.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 54T + 6.89e4T^{2} \) |
| 43 | \( 1 + 20T + 7.95e4T^{2} \) |
| 47 | \( 1 - 394.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 324T + 2.05e5T^{2} \) |
| 61 | \( 1 + 575.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 116T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.10e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 148.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 918T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190T + 9.12e5T^{2} \) |
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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
−9.151764992696714786049703105089, −8.670000418983766598139344983904, −7.930961881339312961116608237921, −6.97373214846031584573255457879, −6.12970043121917535779737568448, −4.63024810956987968438540404625, −3.98815539035158916265374271508, −2.92108406176445060313403921270, −1.59139559536864167804093866819, 0,
1.59139559536864167804093866819, 2.92108406176445060313403921270, 3.98815539035158916265374271508, 4.63024810956987968438540404625, 6.12970043121917535779737568448, 6.97373214846031584573255457879, 7.930961881339312961116608237921, 8.670000418983766598139344983904, 9.151764992696714786049703105089
