| L(s) = 1 | − 3-s − 3·7-s − 2·9-s − 6·13-s − 7·17-s + 5·19-s + 3·21-s − 6·23-s + 5·27-s − 5·29-s + 3·31-s − 3·37-s + 6·39-s − 2·41-s − 4·43-s − 2·47-s + 2·49-s + 7·51-s + 53-s − 5·57-s + 10·59-s − 7·61-s + 6·63-s + 8·67-s + 6·69-s − 7·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.66·13-s − 1.69·17-s + 1.14·19-s + 0.654·21-s − 1.25·23-s + 0.962·27-s − 0.928·29-s + 0.538·31-s − 0.493·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s − 0.291·47-s + 2/7·49-s + 0.980·51-s + 0.137·53-s − 0.662·57-s + 1.30·59-s − 0.896·61-s + 0.755·63-s + 0.977·67-s + 0.722·69-s − 0.830·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−14.92202383992334, −14.14982812113438, −13.84747189872364, −13.23522787435667, −12.69448545579691, −12.20138450620630, −11.74497281754920, −11.34268500799681, −10.69254377995914, −10.00625839012062, −9.728899862571653, −9.199826647298263, −8.576256208040890, −7.921424357782403, −7.257898495815156, −6.736132042758052, −6.334709921691478, −5.667268939919164, −5.106703350609179, −4.621302005773011, −3.772302929577391, −3.167747913456173, −2.456822773116522, −1.957111492248649, −0.5585501371496274, 0,
0.5585501371496274, 1.957111492248649, 2.456822773116522, 3.167747913456173, 3.772302929577391, 4.621302005773011, 5.106703350609179, 5.667268939919164, 6.334709921691478, 6.736132042758052, 7.257898495815156, 7.921424357782403, 8.576256208040890, 9.199826647298263, 9.728899862571653, 10.00625839012062, 10.69254377995914, 11.34268500799681, 11.74497281754920, 12.20138450620630, 12.69448545579691, 13.23522787435667, 13.84747189872364, 14.14982812113438, 14.92202383992334
