| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s + 9-s − 10-s − 12-s − 13-s + 3·14-s − 15-s + 16-s + 3·17-s − 18-s + 4·19-s + 20-s + 3·21-s − 8·23-s + 24-s + 25-s + 26-s − 27-s − 3·28-s − 9·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.654·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s − 1.67·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.77794275997982, −14.45509322878760, −13.75065902039293, −13.23710534059764, −12.65389680483651, −12.22100752457656, −11.82045823625675, −11.08599018420705, −10.63787543525336, −9.978828007194990, −9.755423832230238, −9.216521651389946, −8.778821066020855, −7.725465366695182, −7.453088998660455, −7.020409612177150, −6.007225629374257, −5.894540931995414, −5.487336939548351, −4.390893297645096, −3.780431350529193, −3.112894224113698, −2.369318663611878, −1.668002188385914, −0.7815153769225319, 0,
0.7815153769225319, 1.668002188385914, 2.369318663611878, 3.112894224113698, 3.780431350529193, 4.390893297645096, 5.487336939548351, 5.894540931995414, 6.007225629374257, 7.020409612177150, 7.453088998660455, 7.725465366695182, 8.778821066020855, 9.216521651389946, 9.755423832230238, 9.978828007194990, 10.63787543525336, 11.08599018420705, 11.82045823625675, 12.22100752457656, 12.65389680483651, 13.23710534059764, 13.75065902039293, 14.45509322878760, 14.77794275997982
