| L(s) = 1 | + 4·11-s + 5·13-s + 17-s − 7·19-s − 6·23-s − 5·25-s − 6·29-s − 31-s + 5·37-s + 6·41-s + 43-s − 2·47-s − 6·53-s − 4·59-s − 10·61-s − 13·67-s + 14·71-s + 11·73-s + 5·79-s − 2·83-s + 8·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 1.38·13-s + 0.242·17-s − 1.60·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.179·31-s + 0.821·37-s + 0.937·41-s + 0.152·43-s − 0.291·47-s − 0.824·53-s − 0.520·59-s − 1.28·61-s − 1.58·67-s + 1.66·71-s + 1.28·73-s + 0.562·79-s − 0.219·83-s + 0.847·89-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.80799005873139, −13.35543280986683, −12.83754807648368, −12.37759775720926, −11.85478661398389, −11.36168004239852, −10.91739281058238, −10.56071265037680, −9.837366008669920, −9.292928022899140, −9.040790080275564, −8.397473081405689, −7.830429185188512, −7.585367823473967, −6.546632671513722, −6.255864002017306, −6.049103839760037, −5.308950207155501, −4.428493243869850, −3.975146059397663, −3.762180664215882, −2.970493665062359, −1.961733677003984, −1.774585775999115, −0.8913375140896693, 0,
0.8913375140896693, 1.774585775999115, 1.961733677003984, 2.970493665062359, 3.762180664215882, 3.975146059397663, 4.428493243869850, 5.308950207155501, 6.049103839760037, 6.255864002017306, 6.546632671513722, 7.585367823473967, 7.830429185188512, 8.397473081405689, 9.040790080275564, 9.292928022899140, 9.837366008669920, 10.56071265037680, 10.91739281058238, 11.36168004239852, 11.85478661398389, 12.37759775720926, 12.83754807648368, 13.35543280986683, 13.80799005873139
