| L(s) = 1 | − 2·5-s − 4·7-s + 6·11-s − 13-s + 8·19-s + 6·23-s − 25-s + 8·29-s + 8·35-s + 2·37-s + 4·43-s − 8·47-s + 9·49-s + 6·53-s − 12·55-s − 12·59-s + 8·61-s + 2·65-s − 16·67-s − 8·71-s + 12·73-s − 24·77-s − 10·79-s − 16·83-s + 10·89-s + 4·91-s − 16·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.80·11-s − 0.277·13-s + 1.83·19-s + 1.25·23-s − 1/5·25-s + 1.48·29-s + 1.35·35-s + 0.328·37-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 0.824·53-s − 1.61·55-s − 1.56·59-s + 1.02·61-s + 0.248·65-s − 1.95·67-s − 0.949·71-s + 1.40·73-s − 2.73·77-s − 1.12·79-s − 1.75·83-s + 1.05·89-s + 0.419·91-s − 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.964084482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964084482\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.65545226784474, −12.22422881293828, −11.87507790892993, −11.55500776678159, −11.13339741061099, −10.29747839820272, −10.02780003836523, −9.377020288198173, −9.224999580680213, −8.752226355732340, −8.093061842170719, −7.470935432428962, −7.108975987058547, −6.743945383717074, −6.195754618713807, −5.831668181269084, −5.012176742647082, −4.547304303940333, −3.919905852267827, −3.551767770579516, −2.981952750066281, −2.763529135327556, −1.541358778826411, −1.039119944156591, −0.4498548207304795,
0.4498548207304795, 1.039119944156591, 1.541358778826411, 2.763529135327556, 2.981952750066281, 3.551767770579516, 3.919905852267827, 4.547304303940333, 5.012176742647082, 5.831668181269084, 6.195754618713807, 6.743945383717074, 7.108975987058547, 7.470935432428962, 8.093061842170719, 8.752226355732340, 9.224999580680213, 9.377020288198173, 10.02780003836523, 10.29747839820272, 11.13339741061099, 11.55500776678159, 11.87507790892993, 12.22422881293828, 12.65545226784474
