| L(s) = 1 | − 2.21·2-s − 3-s + 2.90·4-s + 5-s + 2.21·6-s + 3.83·7-s − 2·8-s + 9-s − 2.21·10-s − 5.80·11-s − 2.90·12-s − 8.49·14-s − 15-s − 1.37·16-s − 3.59·17-s − 2.21·18-s − 2.14·19-s + 2.90·20-s − 3.83·21-s + 12.8·22-s − 6.23·23-s + 2·24-s + 25-s − 27-s + 11.1·28-s + 2.06·29-s + 2.21·30-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.45·4-s + 0.447·5-s + 0.903·6-s + 1.45·7-s − 0.707·8-s + 0.333·9-s − 0.700·10-s − 1.75·11-s − 0.838·12-s − 2.27·14-s − 0.258·15-s − 0.344·16-s − 0.871·17-s − 0.521·18-s − 0.492·19-s + 0.649·20-s − 0.837·21-s + 2.74·22-s − 1.30·23-s + 0.408·24-s + 0.200·25-s − 0.192·27-s + 2.10·28-s + 0.383·29-s + 0.404·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6367422122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6367422122\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 5.80T + 11T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 6.23T + 23T^{2} \) |
| 29 | \( 1 - 2.06T + 29T^{2} \) |
| 31 | \( 1 - 5.28T + 31T^{2} \) |
| 37 | \( 1 - 4.75T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 0.815T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 3.55T + 67T^{2} \) |
| 71 | \( 1 + 5.73T + 71T^{2} \) |
| 73 | \( 1 - 2.78T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 0.622T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 4.25T + 97T^{2} \) |
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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
−8.754749692776174649318850495824, −8.068715517982877252595941120847, −7.83031371261817601245944830473, −6.81294370168201225917158220190, −5.96189532717359297550105818682, −5.02297667847651033355559534319, −4.40703275122622177182493974503, −2.45123367125745144890315688980, −1.92600234366391747575454281646, −0.64929413498576609991880651623,
0.64929413498576609991880651623, 1.92600234366391747575454281646, 2.45123367125745144890315688980, 4.40703275122622177182493974503, 5.02297667847651033355559534319, 5.96189532717359297550105818682, 6.81294370168201225917158220190, 7.83031371261817601245944830473, 8.068715517982877252595941120847, 8.754749692776174649318850495824
