| L(s) = 1 | + 5-s − 3·11-s + 13-s + 2·17-s − 19-s + 23-s + 25-s − 2·29-s + 4·31-s − 9·37-s − 3·41-s − 2·43-s + 9·47-s + 9·53-s − 3·55-s − 12·61-s + 65-s + 8·67-s + 16·71-s + 4·73-s − 6·79-s + 4·83-s + 2·85-s − 6·89-s − 95-s + 18·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s + 0.277·13-s + 0.485·17-s − 0.229·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s + 0.718·31-s − 1.47·37-s − 0.468·41-s − 0.304·43-s + 1.31·47-s + 1.23·53-s − 0.404·55-s − 1.53·61-s + 0.124·65-s + 0.977·67-s + 1.89·71-s + 0.468·73-s − 0.675·79-s + 0.439·83-s + 0.216·85-s − 0.635·89-s − 0.102·95-s + 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.59068979532917, −13.29951534360937, −12.62186832800627, −12.28882859760686, −11.83503537281714, −11.07004945527481, −10.80468215547751, −10.18774884545702, −9.960211730182364, −9.299782553515242, −8.693405029725911, −8.426295324833990, −7.730416704597741, −7.313777775027911, −6.741116641004328, −6.172629578393186, −5.682907970335006, −5.075304993541537, −4.852850266371846, −3.850383029060023, −3.553482129541461, −2.701222186209898, −2.324424669848292, −1.569386358768719, −0.8721213927055889, 0,
0.8721213927055889, 1.569386358768719, 2.324424669848292, 2.701222186209898, 3.553482129541461, 3.850383029060023, 4.852850266371846, 5.075304993541537, 5.682907970335006, 6.172629578393186, 6.741116641004328, 7.313777775027911, 7.730416704597741, 8.426295324833990, 8.693405029725911, 9.299782553515242, 9.960211730182364, 10.18774884545702, 10.80468215547751, 11.07004945527481, 11.83503537281714, 12.28882859760686, 12.62186832800627, 13.29951534360937, 13.59068979532917
