| L(s) = 1 | − 2-s + 4-s + 2·5-s − 3·7-s − 8-s − 2·10-s + 5·11-s − 3·13-s + 3·14-s + 16-s + 3·17-s + 5·19-s + 2·20-s − 5·22-s + 3·23-s − 25-s + 3·26-s − 3·28-s + 4·31-s − 32-s − 3·34-s − 6·35-s + 37-s − 5·38-s − 2·40-s + 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.13·7-s − 0.353·8-s − 0.632·10-s + 1.50·11-s − 0.832·13-s + 0.801·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 0.447·20-s − 1.06·22-s + 0.625·23-s − 1/5·25-s + 0.588·26-s − 0.566·28-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.01·35-s + 0.164·37-s − 0.811·38-s − 0.316·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223119248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223119248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.11202593962156360987559406213, −9.479645883044483591577807946944, −9.315757253980273759687218979884, −7.913878554875197130364198555006, −6.87960233611251053044806376508, −6.29485884015171456248050146449, −5.29912923560305098381281180294, −3.69526832182271691380022426077, −2.59442247256705920291581006864, −1.12099772831246061899779950887,
1.12099772831246061899779950887, 2.59442247256705920291581006864, 3.69526832182271691380022426077, 5.29912923560305098381281180294, 6.29485884015171456248050146449, 6.87960233611251053044806376508, 7.913878554875197130364198555006, 9.315757253980273759687218979884, 9.479645883044483591577807946944, 10.11202593962156360987559406213
