| L(s) = 1 | + 2-s + 2·3-s − 4-s − 5-s + 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s − 2·12-s − 2·14-s − 2·15-s − 16-s + 5·17-s + 18-s + 6·19-s + 20-s − 4·21-s + 2·23-s − 6·24-s − 4·25-s − 4·27-s + 2·28-s − 9·29-s − 2·30-s + 2·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s − 0.577·12-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 1.21·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s − 0.872·21-s + 0.417·23-s − 1.22·24-s − 4/5·25-s − 0.769·27-s + 0.377·28-s − 1.67·29-s − 0.365·30-s + 0.359·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.545561844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545561844\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.37635603374626, −14.90090805653959, −14.58662010403816, −13.93002062073324, −13.43977179113303, −13.22548874401430, −12.49764406550026, −11.81510468700938, −11.63577474692643, −10.54558022592982, −9.789410504167825, −9.444224046830106, −9.041899154971132, −8.304968935557827, −7.648998237791104, −7.411938315813778, −6.323587987346882, −5.713149789528768, −5.198847110888931, −4.339524630871453, −3.544639867568488, −3.368043352290190, −2.820202400979652, −1.733284432737575, −0.5422377415320416,
0.5422377415320416, 1.733284432737575, 2.820202400979652, 3.368043352290190, 3.544639867568488, 4.339524630871453, 5.198847110888931, 5.713149789528768, 6.323587987346882, 7.411938315813778, 7.648998237791104, 8.304968935557827, 9.041899154971132, 9.444224046830106, 9.789410504167825, 10.54558022592982, 11.63577474692643, 11.81510468700938, 12.49764406550026, 13.22548874401430, 13.43977179113303, 13.93002062073324, 14.58662010403816, 14.90090805653959, 15.37635603374626
