L(s) = 1 | + 1.08·2-s + 3.06·3-s − 0.830·4-s − 2.59·5-s + 3.30·6-s − 0.985·7-s − 3.06·8-s + 6.36·9-s − 2.80·10-s − 4.60·11-s − 2.54·12-s + 0.224·13-s − 1.06·14-s − 7.93·15-s − 1.64·16-s − 3.99·17-s + 6.88·18-s − 1.95·19-s + 2.15·20-s − 3.01·21-s − 4.97·22-s − 2.30·23-s − 9.36·24-s + 1.72·25-s + 0.242·26-s + 10.3·27-s + 0.819·28-s + ⋯ |
L(s) = 1 | + 0.764·2-s + 1.76·3-s − 0.415·4-s − 1.15·5-s + 1.35·6-s − 0.372·7-s − 1.08·8-s + 2.12·9-s − 0.886·10-s − 1.38·11-s − 0.734·12-s + 0.0622·13-s − 0.284·14-s − 2.04·15-s − 0.412·16-s − 0.968·17-s + 1.62·18-s − 0.447·19-s + 0.481·20-s − 0.658·21-s − 1.06·22-s − 0.480·23-s − 1.91·24-s + 0.344·25-s + 0.0475·26-s + 1.98·27-s + 0.154·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
good | 2 | \( 1 - 1.08T + 2T^{2} \) |
| 3 | \( 1 - 3.06T + 3T^{2} \) |
| 5 | \( 1 + 2.59T + 5T^{2} \) |
| 7 | \( 1 + 0.985T + 7T^{2} \) |
| 11 | \( 1 + 4.60T + 11T^{2} \) |
| 13 | \( 1 - 0.224T + 13T^{2} \) |
| 17 | \( 1 + 3.99T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 - 0.226T + 29T^{2} \) |
| 31 | \( 1 + 0.470T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 - 7.81T + 41T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 - 4.73T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 2.28T + 71T^{2} \) |
| 73 | \( 1 - 8.33T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 5.02T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.609923555958533148871145622218, −8.227182298004847977650378930485, −7.51502828681297998669426643837, −6.62664983163290594287101211947, −5.29210014516821943512398285574, −4.34878332985903017589662482242, −3.80014236873123248254007413372, −3.06123766616228428455248847540, −2.25927272235507279080005076786, 0,
2.25927272235507279080005076786, 3.06123766616228428455248847540, 3.80014236873123248254007413372, 4.34878332985903017589662482242, 5.29210014516821943512398285574, 6.62664983163290594287101211947, 7.51502828681297998669426643837, 8.227182298004847977650378930485, 8.609923555958533148871145622218