| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s − 3·9-s + 2·10-s − 2·11-s + 14-s + 16-s − 17-s + 3·18-s − 2·19-s − 2·20-s + 2·22-s − 8·23-s − 25-s − 28-s + 8·31-s − 32-s + 34-s + 2·35-s − 3·36-s − 4·37-s + 2·38-s + 2·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.632·10-s − 0.603·11-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s − 0.458·19-s − 0.447·20-s + 0.426·22-s − 1.66·23-s − 1/5·25-s − 0.188·28-s + 1.43·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 1/2·36-s − 0.657·37-s + 0.324·38-s + 0.316·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−11.64946522323015128776983368195, −10.67230872441059070869704747981, −9.763847989273936703803471253272, −8.468151783612378810125662561785, −8.024463406411413208815234536452, −6.76626773482443698632812811275, −5.62672058490049396710575997042, −3.96511449610299471402331111472, −2.56493236008968775596152231873, 0,
2.56493236008968775596152231873, 3.96511449610299471402331111472, 5.62672058490049396710575997042, 6.76626773482443698632812811275, 8.024463406411413208815234536452, 8.468151783612378810125662561785, 9.763847989273936703803471253272, 10.67230872441059070869704747981, 11.64946522323015128776983368195
