| L(s) = 1 | − 5.91·3-s − 46.7·5-s + 29.6·7-s − 207.·9-s − 128.·11-s − 83.6·13-s + 276.·15-s + 308.·17-s + 1.12e3·19-s − 175.·21-s − 556.·23-s − 939.·25-s + 2.66e3·27-s + 5.98e3·29-s − 4.58e3·31-s + 760.·33-s − 1.38e3·35-s + 9.19e3·37-s + 495.·39-s − 5.49e3·41-s + 8.28e3·43-s + 9.72e3·45-s + 2.22e4·47-s − 1.59e4·49-s − 1.82e3·51-s + 9.63e3·53-s + 6.00e3·55-s + ⋯ |
| L(s) = 1 | − 0.379·3-s − 0.836·5-s + 0.229·7-s − 0.855·9-s − 0.320·11-s − 0.137·13-s + 0.317·15-s + 0.258·17-s + 0.717·19-s − 0.0869·21-s − 0.219·23-s − 0.300·25-s + 0.704·27-s + 1.32·29-s − 0.856·31-s + 0.121·33-s − 0.191·35-s + 1.10·37-s + 0.0521·39-s − 0.510·41-s + 0.683·43-s + 0.715·45-s + 1.46·47-s − 0.947·49-s − 0.0982·51-s + 0.471·53-s + 0.267·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 5.91T + 243T^{2} \) |
| 5 | \( 1 + 46.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 29.6T + 1.68e4T^{2} \) |
| 11 | \( 1 + 128.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 83.6T + 3.71e5T^{2} \) |
| 17 | \( 1 - 308.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 556.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.98e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.19e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.49e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.22e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.63e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.95e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.61e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.06e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.24e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.04e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.61e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.43e5T + 8.58e9T^{2} \) |
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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
−8.653292371750467755809639676362, −7.940904456265203428770367006249, −7.26154878690653093909019397457, −6.13315933301630376238320512373, −5.36088935489321021410828961183, −4.45365859101053923131550392118, −3.42269983932151110447639535270, −2.46725855761929332569090213579, −0.963166217514519470848231293389, 0,
0.963166217514519470848231293389, 2.46725855761929332569090213579, 3.42269983932151110447639535270, 4.45365859101053923131550392118, 5.36088935489321021410828961183, 6.13315933301630376238320512373, 7.26154878690653093909019397457, 7.940904456265203428770367006249, 8.653292371750467755809639676362
