| L(s) = 1 | + 14·3-s + 158·7-s − 47·9-s + 148·11-s − 684·13-s − 2.04e3·17-s − 2.22e3·19-s + 2.21e3·21-s − 1.24e3·23-s − 4.06e3·27-s − 270·29-s + 2.04e3·31-s + 2.07e3·33-s + 4.37e3·37-s − 9.57e3·39-s − 2.39e3·41-s + 2.29e3·43-s − 1.06e4·47-s + 8.15e3·49-s − 2.86e4·51-s − 2.96e3·53-s − 3.10e4·57-s + 3.97e4·59-s − 4.22e4·61-s − 7.42e3·63-s + 3.20e4·67-s − 1.74e4·69-s + ⋯ |
| L(s) = 1 | + 0.898·3-s + 1.21·7-s − 0.193·9-s + 0.368·11-s − 1.12·13-s − 1.71·17-s − 1.41·19-s + 1.09·21-s − 0.491·23-s − 1.07·27-s − 0.0596·29-s + 0.382·31-s + 0.331·33-s + 0.525·37-s − 1.00·39-s − 0.222·41-s + 0.189·43-s − 0.705·47-s + 0.485·49-s − 1.54·51-s − 0.144·53-s − 1.26·57-s + 1.48·59-s − 1.45·61-s − 0.235·63-s + 0.873·67-s − 0.441·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 - 158 T + p^{5} T^{2} \) |
| 11 | \( 1 - 148 T + p^{5} T^{2} \) |
| 13 | \( 1 + 684 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2048 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2220 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1246 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2048 T + p^{5} T^{2} \) |
| 37 | \( 1 - 4372 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2398 T + p^{5} T^{2} \) |
| 43 | \( 1 - 2294 T + p^{5} T^{2} \) |
| 47 | \( 1 + 10682 T + p^{5} T^{2} \) |
| 53 | \( 1 + 2964 T + p^{5} T^{2} \) |
| 59 | \( 1 - 39740 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42298 T + p^{5} T^{2} \) |
| 67 | \( 1 - 32098 T + p^{5} T^{2} \) |
| 71 | \( 1 - 4248 T + p^{5} T^{2} \) |
| 73 | \( 1 + 30104 T + p^{5} T^{2} \) |
| 79 | \( 1 + 35280 T + p^{5} T^{2} \) |
| 83 | \( 1 + 27826 T + p^{5} T^{2} \) |
| 89 | \( 1 + 85210 T + p^{5} T^{2} \) |
| 97 | \( 1 - 97232 T + p^{5} T^{2} \) |
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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
−9.884620709906532758768957842059, −8.817228191598518514928911629022, −8.352345148345548116621176433427, −7.39943138420528753341728918011, −6.27855085331064434988939796234, −4.86364865258870609673186413208, −4.10701862125027924524668219856, −2.54371976867165853289592280491, −1.86816374927412047878431731771, 0,
1.86816374927412047878431731771, 2.54371976867165853289592280491, 4.10701862125027924524668219856, 4.86364865258870609673186413208, 6.27855085331064434988939796234, 7.39943138420528753341728918011, 8.352345148345548116621176433427, 8.817228191598518514928911629022, 9.884620709906532758768957842059
