| L(s) = 1 | + 5-s − 4·11-s − 13-s + 3·17-s − 6·19-s + 9·23-s + 25-s − 6·31-s − 6·37-s − 5·41-s + 6·43-s + 3·53-s − 4·55-s + 13·59-s + 5·61-s − 65-s + 5·67-s + 71-s + 7·73-s − 6·83-s + 3·85-s − 14·89-s − 6·95-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.277·13-s + 0.727·17-s − 1.37·19-s + 1.87·23-s + 1/5·25-s − 1.07·31-s − 0.986·37-s − 0.780·41-s + 0.914·43-s + 0.412·53-s − 0.539·55-s + 1.69·59-s + 0.640·61-s − 0.124·65-s + 0.610·67-s + 0.118·71-s + 0.819·73-s − 0.658·83-s + 0.325·85-s − 1.48·89-s − 0.615·95-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.15545144795795, −12.80040169593141, −12.37450694255986, −11.82838041237620, −11.11018542608526, −10.79021182528222, −10.50080998552088, −9.868913684597202, −9.548421487457864, −8.849427337025481, −8.497189669918066, −8.094268525094860, −7.337156580479957, −7.021707496569215, −6.630889377541133, −5.750142172884948, −5.495023379478554, −5.034336471271853, −4.538391352342511, −3.747908516963674, −3.313477188817800, −2.540380949504762, −2.290855756991801, −1.515591824715722, −0.7678692564297166, 0,
0.7678692564297166, 1.515591824715722, 2.290855756991801, 2.540380949504762, 3.313477188817800, 3.747908516963674, 4.538391352342511, 5.034336471271853, 5.495023379478554, 5.750142172884948, 6.630889377541133, 7.021707496569215, 7.337156580479957, 8.094268525094860, 8.497189669918066, 8.849427337025481, 9.548421487457864, 9.868913684597202, 10.50080998552088, 10.79021182528222, 11.11018542608526, 11.82838041237620, 12.37450694255986, 12.80040169593141, 13.15545144795795
