| L(s) = 1 | − 2-s + 3.09·3-s + 4-s − 5-s − 3.09·6-s − 2.30·7-s − 8-s + 6.56·9-s + 10-s − 11-s + 3.09·12-s + 1.84·13-s + 2.30·14-s − 3.09·15-s + 16-s − 1.97·17-s − 6.56·18-s − 6.24·19-s − 20-s − 7.14·21-s + 22-s − 1.04·23-s − 3.09·24-s + 25-s − 1.84·26-s + 11.0·27-s − 2.30·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·3-s + 0.5·4-s − 0.447·5-s − 1.26·6-s − 0.872·7-s − 0.353·8-s + 2.18·9-s + 0.316·10-s − 0.301·11-s + 0.892·12-s + 0.510·13-s + 0.617·14-s − 0.798·15-s + 0.250·16-s − 0.480·17-s − 1.54·18-s − 1.43·19-s − 0.223·20-s − 1.55·21-s + 0.213·22-s − 0.218·23-s − 0.631·24-s + 0.200·25-s − 0.361·26-s + 2.12·27-s − 0.436·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 3.09T + 3T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 + 6.24T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 5.26T + 29T^{2} \) |
| 31 | \( 1 - 4.56T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 1.67T + 41T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 + 8.57T + 61T^{2} \) |
| 67 | \( 1 + 8.29T + 67T^{2} \) |
| 71 | \( 1 - 8.71T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 7.00T + 83T^{2} \) |
| 89 | \( 1 - 2.89T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.151471659452174442600727760160, −7.44036027042329818224839474570, −6.78853386264329802883151976612, −6.10703802109948934764112563285, −4.66943486877229852501837361769, −3.78769908828080848754312561171, −3.24864342208119081464794846175, −2.41400060245662222865738852388, −1.63185351077766238386434785144, 0,
1.63185351077766238386434785144, 2.41400060245662222865738852388, 3.24864342208119081464794846175, 3.78769908828080848754312561171, 4.66943486877229852501837361769, 6.10703802109948934764112563285, 6.78853386264329802883151976612, 7.44036027042329818224839474570, 8.151471659452174442600727760160
