| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s − 2·9-s + 10-s − 3·11-s + 12-s + 2·13-s + 14-s − 15-s + 16-s + 3·17-s + 2·18-s − 8·19-s − 20-s − 21-s + 3·22-s − 24-s + 25-s − 2·26-s − 5·27-s − 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.83·19-s − 0.223·20-s − 0.218·21-s + 0.639·22-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.962·27-s − 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−15.42530897983693, −14.75878804632036, −14.66931180573404, −13.92311265253918, −13.03490297161217, −12.95524440996888, −12.34812430396983, −11.47181518307471, −11.15311271429512, −10.62007288899053, −10.15421981309205, −9.317218763425716, −9.035891876146024, −8.439708858909075, −7.816928420815960, −7.713426534631875, −6.778531365515605, −6.214081017595002, −5.632284086725865, −4.976307110954100, −4.006439260221334, −3.437053756174753, −2.943472811771963, −2.128415168321304, −1.543234380220344, 0, 0,
1.543234380220344, 2.128415168321304, 2.943472811771963, 3.437053756174753, 4.006439260221334, 4.976307110954100, 5.632284086725865, 6.214081017595002, 6.778531365515605, 7.713426534631875, 7.816928420815960, 8.439708858909075, 9.035891876146024, 9.317218763425716, 10.15421981309205, 10.62007288899053, 11.15311271429512, 11.47181518307471, 12.34812430396983, 12.95524440996888, 13.03490297161217, 13.92311265253918, 14.66931180573404, 14.75878804632036, 15.42530897983693
