| L(s) = 1 | − 3-s − 2·4-s + 5-s − 7-s − 2·9-s − 3·11-s + 2·12-s + 5·13-s − 15-s + 4·16-s + 3·17-s − 2·19-s − 2·20-s + 21-s − 6·23-s + 25-s + 5·27-s + 2·28-s − 3·29-s − 4·31-s + 3·33-s − 35-s + 4·36-s − 2·37-s − 5·39-s − 12·41-s + 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 1.38·13-s − 0.258·15-s + 16-s + 0.727·17-s − 0.458·19-s − 0.447·20-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.522·33-s − 0.169·35-s + 2/3·36-s − 0.328·37-s − 0.800·39-s − 1.87·41-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.42178425408011, −13.77106134627716, −13.41154403955116, −13.24592317262737, −12.30843983512943, −12.19334958969421, −11.47059735044065, −10.75229054108869, −10.35186940376949, −10.14634363325781, −9.234157248848498, −8.907781666521111, −8.370797614731858, −7.922897701276733, −7.241979448862465, −6.374963563052200, −5.957252729482368, −5.523089934493860, −5.168409759743318, −4.366185468954585, −3.638744415915559, −3.325929390910893, −2.393134444813448, −1.596175382999599, −0.6945055441275580, 0,
0.6945055441275580, 1.596175382999599, 2.393134444813448, 3.325929390910893, 3.638744415915559, 4.366185468954585, 5.168409759743318, 5.523089934493860, 5.957252729482368, 6.374963563052200, 7.241979448862465, 7.922897701276733, 8.370797614731858, 8.907781666521111, 9.234157248848498, 10.14634363325781, 10.35186940376949, 10.75229054108869, 11.47059735044065, 12.19334958969421, 12.30843983512943, 13.24592317262737, 13.41154403955116, 13.77106134627716, 14.42178425408011
