| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·11-s − 4·13-s + 15-s − 7·17-s + 2·21-s + 9·23-s − 4·25-s − 27-s + 10·29-s − 8·31-s + 2·33-s + 2·35-s + 7·37-s + 4·39-s + 12·41-s + 43-s − 45-s + 8·47-s − 3·49-s + 7·51-s + 6·53-s + 2·55-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s − 1.69·17-s + 0.436·21-s + 1.87·23-s − 4/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s + 0.348·33-s + 0.338·35-s + 1.15·37-s + 0.640·39-s + 1.87·41-s + 0.152·43-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.980·51-s + 0.824·53-s + 0.269·55-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.54548736978077, −16.17231416335470, −15.59126577592131, −15.15665078981563, −14.57665384692272, −13.68946758891946, −13.15112828558070, −12.64529796485655, −12.25765778489506, −11.41920244721374, −10.92519493164863, −10.53709685115530, −9.552964103316364, −9.329705657269652, −8.486723291762220, −7.678215888006047, −7.089769154403436, −6.651533737162745, −5.870585849065852, −5.114021467328515, −4.524158556060043, −3.885417325427622, −2.753601225238413, −2.377627196965808, −0.8653175994650593, 0,
0.8653175994650593, 2.377627196965808, 2.753601225238413, 3.885417325427622, 4.524158556060043, 5.114021467328515, 5.870585849065852, 6.651533737162745, 7.089769154403436, 7.678215888006047, 8.486723291762220, 9.329705657269652, 9.552964103316364, 10.53709685115530, 10.92519493164863, 11.41920244721374, 12.25765778489506, 12.64529796485655, 13.15112828558070, 13.68946758891946, 14.57665384692272, 15.15665078981563, 15.59126577592131, 16.17231416335470, 16.54548736978077
