L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 11-s + 3·13-s + 15-s + 17-s − 5·19-s − 21-s + 7·23-s − 4·25-s + 27-s + 6·29-s + 4·31-s + 33-s − 35-s + 2·37-s + 3·39-s − 3·41-s + 7·43-s + 45-s − 4·47-s + 49-s + 51-s + 55-s − 5·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.832·13-s + 0.258·15-s + 0.242·17-s − 1.14·19-s − 0.218·21-s + 1.45·23-s − 4/5·25-s + 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s − 0.169·35-s + 0.328·37-s + 0.480·39-s − 0.468·41-s + 1.06·43-s + 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.140·51-s + 0.134·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.450096824\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.450096824\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−15.60738179711590, −14.78885316992937, −14.51704477704421, −13.75918224789688, −13.36731573934866, −12.96337683596087, −12.32104577261678, −11.72696945960596, −10.97350004017310, −10.56213253342330, −9.850677415231462, −9.465131952107521, −8.679363400392526, −8.469460162572450, −7.709326015781265, −6.902025685257077, −6.467671654260723, −5.918349284386895, −5.122212304165880, −4.354909081633755, −3.786168100481584, −3.012559452657830, −2.431800163124788, −1.550333723950236, −0.7546684011825471,
0.7546684011825471, 1.550333723950236, 2.431800163124788, 3.012559452657830, 3.786168100481584, 4.354909081633755, 5.122212304165880, 5.918349284386895, 6.467671654260723, 6.902025685257077, 7.709326015781265, 8.469460162572450, 8.679363400392526, 9.465131952107521, 9.850677415231462, 10.56213253342330, 10.97350004017310, 11.72696945960596, 12.32104577261678, 12.96337683596087, 13.36731573934866, 13.75918224789688, 14.51704477704421, 14.78885316992937, 15.60738179711590