| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 4·11-s − 12-s − 6·13-s + 14-s + 16-s − 18-s − 4·19-s + 21-s − 4·22-s + 8·23-s + 24-s + 6·26-s − 27-s − 28-s + 2·29-s − 32-s − 4·33-s + 36-s − 10·37-s + 4·38-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 + 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s − 0.188·28-s + 0.371·29-s − 0.176·32-s − 0.696·33-s + 1/6·36-s − 1.64·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−12.69076105656652, −12.32077366403183, −12.04668874885997, −11.51038400512981, −10.96867892266566, −10.63765542790003, −10.18843373564795, −9.584804008123720, −9.293644268546808, −8.933091522845267, −8.367888551931455, −7.718371337744722, −7.231237376826510, −6.734304810794846, −6.677748451078704, −5.910167069751624, −5.412106798921566, −4.807011936184105, −4.403622113233038, −3.777976656089874, −3.061988343122616, −2.613503091332836, −1.910986502099330, −1.336770019474057, −0.6451709050847912, 0,
0.6451709050847912, 1.336770019474057, 1.910986502099330, 2.613503091332836, 3.061988343122616, 3.777976656089874, 4.403622113233038, 4.807011936184105, 5.412106798921566, 5.910167069751624, 6.677748451078704, 6.734304810794846, 7.231237376826510, 7.718371337744722, 8.367888551931455, 8.933091522845267, 9.293644268546808, 9.584804008123720, 10.18843373564795, 10.63765542790003, 10.96867892266566, 11.51038400512981, 12.04668874885997, 12.32077366403183, 12.69076105656652
