| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 3·11-s + 13-s + 15-s − 3·17-s − 6·19-s − 3·21-s − 4·23-s + 25-s + 27-s + 29-s − 4·31-s − 3·33-s − 3·35-s − 4·37-s + 39-s + 2·41-s + 4·43-s + 45-s − 3·47-s + 2·49-s − 3·51-s + 6·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 1.37·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s − 0.718·31-s − 0.522·33-s − 0.507·35-s − 0.657·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.420·51-s + 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−8.977772545976198663458639648741, −8.248029066882421296740697658734, −7.33548636345869317829741092214, −6.45094562380598595733597865632, −5.87968209883743333563161571072, −4.68670697707933784822821040344, −3.73949324550628996073299112294, −2.77551981441276545461339322850, −1.94725580514987939336132871333, 0,
1.94725580514987939336132871333, 2.77551981441276545461339322850, 3.73949324550628996073299112294, 4.68670697707933784822821040344, 5.87968209883743333563161571072, 6.45094562380598595733597865632, 7.33548636345869317829741092214, 8.248029066882421296740697658734, 8.977772545976198663458639648741
