| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 15-s + 16-s − 18-s − 19-s − 20-s + 24-s + 25-s − 27-s − 30-s − 32-s + 36-s + 38-s + 40-s − 45-s − 48-s − 49-s − 50-s + 2·53-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s + 15-s + 16-s − 18-s − 19-s − 20-s + 24-s + 25-s − 27-s − 30-s − 32-s + 36-s + 38-s + 40-s − 45-s − 48-s − 49-s − 50-s + 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3870215290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3870215290\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−9.155171962430298048597364167184, −8.392850450159217094121106835052, −7.67900183073201085108217941061, −6.94117271501523476540784011378, −6.36261615833280163332555866641, −5.39644911830679853444687875443, −4.39258961813203148241749345082, −3.46382345358157202994862339452, −2.09462370975376312751494933779, −0.71370802561562918809958745811,
0.71370802561562918809958745811, 2.09462370975376312751494933779, 3.46382345358157202994862339452, 4.39258961813203148241749345082, 5.39644911830679853444687875443, 6.36261615833280163332555866641, 6.94117271501523476540784011378, 7.67900183073201085108217941061, 8.392850450159217094121106835052, 9.155171962430298048597364167184
