| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 6·13-s + 15-s − 2·17-s − 4·19-s + 21-s + 23-s + 25-s − 27-s + 6·29-s − 4·31-s + 35-s + 2·37-s + 6·39-s + 10·41-s − 8·43-s − 45-s − 4·47-s + 49-s + 2·51-s + 10·53-s + 4·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 0.960·39-s + 1.56·41-s − 1.21·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.92688728684563, −14.76980068591040, −14.17833263717739, −13.28266191044522, −12.96816375246182, −12.45678725938908, −11.88879143734239, −11.59117138115106, −10.86603456557383, −10.25609502045708, −10.05741643285364, −9.139318442774472, −8.868391809978921, −7.992674296619199, −7.465059517455510, −6.992024934702175, −6.409063730375024, −5.867064024993047, −5.041686680635383, −4.614963443904834, −4.109919610177957, −3.216390286910926, −2.544898668574989, −1.882040258759358, −0.7191116070452875, 0,
0.7191116070452875, 1.882040258759358, 2.544898668574989, 3.216390286910926, 4.109919610177957, 4.614963443904834, 5.041686680635383, 5.867064024993047, 6.409063730375024, 6.992024934702175, 7.465059517455510, 7.992674296619199, 8.868391809978921, 9.139318442774472, 10.05741643285364, 10.25609502045708, 10.86603456557383, 11.59117138115106, 11.88879143734239, 12.45678725938908, 12.96816375246182, 13.28266191044522, 14.17833263717739, 14.76980068591040, 14.92688728684563
