| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s + 5·13-s − 5·17-s − 4·19-s + 21-s − 7·23-s − 27-s + 3·29-s − 31-s + 33-s + 8·37-s − 5·39-s − 11·41-s − 43-s − 2·47-s + 49-s + 5·51-s + 3·53-s + 4·57-s − 11·59-s − 7·61-s − 63-s − 12·67-s + 7·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s − 1.21·17-s − 0.917·19-s + 0.218·21-s − 1.45·23-s − 0.192·27-s + 0.557·29-s − 0.179·31-s + 0.174·33-s + 1.31·37-s − 0.800·39-s − 1.71·41-s − 0.152·43-s − 0.291·47-s + 1/7·49-s + 0.700·51-s + 0.412·53-s + 0.529·57-s − 1.43·59-s − 0.896·61-s − 0.125·63-s − 1.46·67-s + 0.842·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5039822248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5039822248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.53234276095881, −11.98067241347230, −11.61863240279095, −11.05207064369183, −10.75951467262027, −10.37837197067318, −9.823021234441239, −9.419334246061864, −8.680486720230005, −8.530510424333404, −7.979229351680153, −7.444010149741775, −6.734031366965739, −6.453865447539669, −6.013785450198262, −5.751438619194852, −4.877051627033289, −4.506254794650317, −4.051866159365405, −3.496900037177142, −2.941030032592586, −2.150816720361691, −1.783497390377013, −1.035542057480887, −0.2051082377176234,
0.2051082377176234, 1.035542057480887, 1.783497390377013, 2.150816720361691, 2.941030032592586, 3.496900037177142, 4.051866159365405, 4.506254794650317, 4.877051627033289, 5.751438619194852, 6.013785450198262, 6.453865447539669, 6.734031366965739, 7.444010149741775, 7.979229351680153, 8.530510424333404, 8.680486720230005, 9.419334246061864, 9.823021234441239, 10.37837197067318, 10.75951467262027, 11.05207064369183, 11.61863240279095, 11.98067241347230, 12.53234276095881
