| L(s) = 1 | + 3·3-s − 4·5-s + 7-s + 6·9-s + 5·13-s − 12·15-s − 5·17-s + 3·21-s − 3·23-s + 11·25-s + 9·27-s − 7·29-s + 10·31-s − 4·35-s + 2·37-s + 15·39-s − 6·41-s − 4·43-s − 24·45-s − 8·47-s − 6·49-s − 15·51-s + 9·53-s − 59-s − 2·61-s + 6·63-s − 20·65-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 0.377·7-s + 2·9-s + 1.38·13-s − 3.09·15-s − 1.21·17-s + 0.654·21-s − 0.625·23-s + 11/5·25-s + 1.73·27-s − 1.29·29-s + 1.79·31-s − 0.676·35-s + 0.328·37-s + 2.40·39-s − 0.937·41-s − 0.609·43-s − 3.57·45-s − 1.16·47-s − 6/7·49-s − 2.10·51-s + 1.23·53-s − 0.130·59-s − 0.256·61-s + 0.755·63-s − 2.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−14.14306167557501, −13.50034646466073, −13.29768118213743, −12.79544831771733, −12.07038742656032, −11.56937898879618, −11.23736622653849, −10.69639680537039, −10.03815802109831, −9.445708029171518, −8.810608352970011, −8.465244952363844, −8.081833097371399, −7.917803565577034, −7.092121184561864, −6.762660579562605, −6.092135142395941, −5.000963931038090, −4.473383335718264, −4.011830089966114, −3.565045201575187, −3.189177602148249, −2.412377910729495, −1.759817872365039, −1.011246593498862, 0,
1.011246593498862, 1.759817872365039, 2.412377910729495, 3.189177602148249, 3.565045201575187, 4.011830089966114, 4.473383335718264, 5.000963931038090, 6.092135142395941, 6.762660579562605, 7.092121184561864, 7.917803565577034, 8.081833097371399, 8.465244952363844, 8.810608352970011, 9.445708029171518, 10.03815802109831, 10.69639680537039, 11.23736622653849, 11.56937898879618, 12.07038742656032, 12.79544831771733, 13.29768118213743, 13.50034646466073, 14.14306167557501
