| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 11-s − 12-s − 14-s − 15-s + 16-s + 17-s + 19-s + 20-s − 21-s + 22-s + 24-s + 25-s + 27-s + 28-s + 29-s + 30-s − 31-s − 32-s + 33-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 10-s − 11-s − 12-s − 14-s − 15-s + 16-s + 17-s + 19-s + 20-s − 21-s + 22-s + 24-s + 25-s + 27-s + 28-s + 29-s + 30-s − 31-s − 32-s + 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5164202472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5164202472\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( ( 1 - T )^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 + T )^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.12204549040990683717516150981, −10.45067892184719181627143272740, −9.769786204691378467800566710505, −8.625913659267304926393364957720, −7.76762428351337135778608708561, −6.72973880701448080455695617179, −5.55111905902122009946267710020, −5.22365011277454663764537482709, −2.84947704920727500924204999401, −1.40622344528720118735173532471,
1.40622344528720118735173532471, 2.84947704920727500924204999401, 5.22365011277454663764537482709, 5.55111905902122009946267710020, 6.72973880701448080455695617179, 7.76762428351337135778608708561, 8.625913659267304926393364957720, 9.769786204691378467800566710505, 10.45067892184719181627143272740, 11.12204549040990683717516150981
