| L(s) = 1 | + 0.219·3-s − 0.732·5-s + 2.55·7-s − 2.95·9-s + 4.06·11-s − 0.307·13-s − 0.160·15-s − 6.15·17-s − 2.57·19-s + 0.560·21-s − 5.93·23-s − 4.46·25-s − 1.30·27-s + 4.60·29-s + 1.33·31-s + 0.892·33-s − 1.86·35-s − 2.91·37-s − 0.0675·39-s − 3.83·41-s − 5.80·43-s + 2.16·45-s − 47-s − 0.487·49-s − 1.35·51-s + 5.68·53-s − 2.97·55-s + ⋯ |
| L(s) = 1 | + 0.126·3-s − 0.327·5-s + 0.964·7-s − 0.983·9-s + 1.22·11-s − 0.0852·13-s − 0.0415·15-s − 1.49·17-s − 0.589·19-s + 0.122·21-s − 1.23·23-s − 0.892·25-s − 0.251·27-s + 0.854·29-s + 0.239·31-s + 0.155·33-s − 0.315·35-s − 0.478·37-s − 0.0108·39-s − 0.599·41-s − 0.884·43-s + 0.322·45-s − 0.145·47-s − 0.0696·49-s − 0.189·51-s + 0.781·53-s − 0.401·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 - 0.219T + 3T^{2} \) |
| 5 | \( 1 + 0.732T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 4.06T + 11T^{2} \) |
| 13 | \( 1 + 0.307T + 13T^{2} \) |
| 17 | \( 1 + 6.15T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 1.33T + 31T^{2} \) |
| 37 | \( 1 + 2.91T + 37T^{2} \) |
| 41 | \( 1 + 3.83T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 + 3.08T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 + 1.63T + 71T^{2} \) |
| 73 | \( 1 - 3.61T + 73T^{2} \) |
| 79 | \( 1 - 3.59T + 79T^{2} \) |
| 83 | \( 1 - 6.74T + 83T^{2} \) |
| 89 | \( 1 - 7.28T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
| 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
−8.457614639464161485668883147968, −7.80536640832325714578193372007, −6.71401948148617095401124155187, −6.22196194367857802007854175569, −5.18878048783614271264319522329, −4.34054278366765581641735758381, −3.73253724020968252152547817620, −2.45939837720047793115201086558, −1.63082561647260745508180359554, 0,
1.63082561647260745508180359554, 2.45939837720047793115201086558, 3.73253724020968252152547817620, 4.34054278366765581641735758381, 5.18878048783614271264319522329, 6.22196194367857802007854175569, 6.71401948148617095401124155187, 7.80536640832325714578193372007, 8.457614639464161485668883147968
