| L(s) = 1 | − 3-s + 4·5-s + 9-s − 2·11-s + 13-s − 4·15-s − 6·17-s + 19-s − 23-s + 11·25-s − 27-s − 10·29-s − 7·31-s + 2·33-s + 3·37-s − 39-s − 12·41-s + 7·43-s + 4·45-s − 10·47-s + 6·51-s − 12·53-s − 8·55-s − 57-s − 14·59-s − 10·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.03·15-s − 1.45·17-s + 0.229·19-s − 0.208·23-s + 11/5·25-s − 0.192·27-s − 1.85·29-s − 1.25·31-s + 0.348·33-s + 0.493·37-s − 0.160·39-s − 1.87·41-s + 1.06·43-s + 0.596·45-s − 1.45·47-s + 0.840·51-s − 1.64·53-s − 1.07·55-s − 0.132·57-s − 1.82·59-s − 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7797123856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7797123856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.11192238181509, −12.71682117361726, −12.14332027559088, −11.37341958864629, −11.05585369077896, −10.69104587191937, −10.25665957251429, −9.616216108511126, −9.316446873609132, −9.013380449444968, −8.312914606428130, −7.684295725972360, −7.133302078547225, −6.644526177659936, −6.096168909779826, −5.850831309287565, −5.320406331678279, −4.808485034109311, −4.404651911974681, −3.480082871598317, −2.989301436011019, −2.217112325238917, −1.716831749141524, −1.488542824231972, −0.2315158910432452,
0.2315158910432452, 1.488542824231972, 1.716831749141524, 2.217112325238917, 2.989301436011019, 3.480082871598317, 4.404651911974681, 4.808485034109311, 5.320406331678279, 5.850831309287565, 6.096168909779826, 6.644526177659936, 7.133302078547225, 7.684295725972360, 8.312914606428130, 9.013380449444968, 9.316446873609132, 9.616216108511126, 10.25665957251429, 10.69104587191937, 11.05585369077896, 11.37341958864629, 12.14332027559088, 12.71682117361726, 13.11192238181509
