| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 11-s + 12-s + 4·13-s + 14-s + 16-s + 8·17-s + 18-s − 19-s + 21-s + 22-s + 4·23-s + 24-s + 4·26-s + 27-s + 28-s + 10·29-s − 8·31-s + 32-s + 33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 0.229·19-s + 0.218·21-s + 0.213·22-s + 0.834·23-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.188·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.174·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.28506959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.28506959\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.93962841706237, −12.65912070527881, −11.97836757972630, −11.84947641267748, −11.03101545779613, −10.72162864816019, −10.35283403702091, −9.568900666832866, −9.315757445141451, −8.555749455167317, −8.322696166813866, −7.627305444670828, −7.383972991877556, −6.721659143795887, −6.142812175445973, −5.710029416230423, −5.267249128435133, −4.532945957812970, −4.158951914210846, −3.536681662115235, −3.140483131554420, −2.602267967839479, −1.891222844578449, −1.080454373390682, −0.9330100965750929,
0.9330100965750929, 1.080454373390682, 1.891222844578449, 2.602267967839479, 3.140483131554420, 3.536681662115235, 4.158951914210846, 4.532945957812970, 5.267249128435133, 5.710029416230423, 6.142812175445973, 6.721659143795887, 7.383972991877556, 7.627305444670828, 8.322696166813866, 8.555749455167317, 9.315757445141451, 9.568900666832866, 10.35283403702091, 10.72162864816019, 11.03101545779613, 11.84947641267748, 11.97836757972630, 12.65912070527881, 12.93962841706237
