| L(s) = 1 | + 5.03·3-s − 40.9·5-s − 196.·7-s − 217.·9-s + 679.·11-s − 975.·13-s − 206.·15-s + 1.21e3·17-s + 955.·19-s − 989.·21-s + 2.13e3·23-s − 1.44e3·25-s − 2.32e3·27-s + 7.74e3·29-s + 8.76e3·31-s + 3.42e3·33-s + 8.03e3·35-s − 1.01e3·37-s − 4.91e3·39-s − 6.34e3·41-s − 1.31e4·43-s + 8.91e3·45-s + 5.83e3·47-s + 2.17e4·49-s + 6.11e3·51-s + 9.56e3·53-s − 2.78e4·55-s + ⋯ |
| L(s) = 1 | + 0.323·3-s − 0.732·5-s − 1.51·7-s − 0.895·9-s + 1.69·11-s − 1.60·13-s − 0.236·15-s + 1.01·17-s + 0.607·19-s − 0.489·21-s + 0.843·23-s − 0.463·25-s − 0.612·27-s + 1.71·29-s + 1.63·31-s + 0.547·33-s + 1.10·35-s − 0.122·37-s − 0.517·39-s − 0.589·41-s − 1.08·43-s + 0.655·45-s + 0.385·47-s + 1.29·49-s + 0.329·51-s + 0.467·53-s − 1.23·55-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 | \( 1 - 5.03T + 243T^{2} \) |
| 5 | \( 1 + 40.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 196.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 679.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 975.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.21e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 955.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.13e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.74e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.34e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.31e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.83e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.88e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.69e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.01e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.07e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.20e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.97e4T + 8.58e9T^{2} \) |
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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
−8.836266696902666861876398692482, −8.019145276107764385785658115372, −6.97971636139397804244497643308, −6.49072430462706775341618817977, −5.35656494328713431810149913984, −4.20283210641384220630487320466, −3.24788897377497783545727614523, −2.75901399942702193726479643948, −1.00915174261996199343953152198, 0,
1.00915174261996199343953152198, 2.75901399942702193726479643948, 3.24788897377497783545727614523, 4.20283210641384220630487320466, 5.35656494328713431810149913984, 6.49072430462706775341618817977, 6.97971636139397804244497643308, 8.019145276107764385785658115372, 8.836266696902666861876398692482
