| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s − 4·11-s − 6·13-s − 15-s + 6·17-s + 21-s + 4·23-s + 25-s + 27-s − 6·29-s − 4·33-s − 35-s + 2·37-s − 6·39-s + 2·41-s − 4·43-s − 45-s − 4·47-s + 49-s + 6·51-s − 6·53-s + 4·55-s − 12·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 0.258·15-s + 1.45·17-s + 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s − 0.583·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s + 0.539·55-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.975857558819534712992785983566, −7.63842307329062980786406153706, −7.15949706317031859917028175740, −5.84299976937437989512602583264, −5.04635759900256161424971300272, −4.50205207087059837220327938148, −3.24635687793267322096048194020, −2.74009819930328431677003644600, −1.58256316454307986080035448088, 0,
1.58256316454307986080035448088, 2.74009819930328431677003644600, 3.24635687793267322096048194020, 4.50205207087059837220327938148, 5.04635759900256161424971300272, 5.84299976937437989512602583264, 7.15949706317031859917028175740, 7.63842307329062980786406153706, 7.975857558819534712992785983566
