| L(s) = 1 | − 2.45·2-s − 3-s + 4.02·4-s + 5-s + 2.45·6-s − 4.96·8-s + 9-s − 2.45·10-s − 4.49·11-s − 4.02·12-s − 13-s − 15-s + 4.13·16-s − 2.69·17-s − 2.45·18-s − 4.19·19-s + 4.02·20-s + 11.0·22-s − 4.55·23-s + 4.96·24-s + 25-s + 2.45·26-s − 27-s − 8.15·29-s + 2.45·30-s − 8.41·31-s − 0.224·32-s + ⋯ |
| L(s) = 1 | − 1.73·2-s − 0.577·3-s + 2.01·4-s + 0.447·5-s + 1.00·6-s − 1.75·8-s + 0.333·9-s − 0.776·10-s − 1.35·11-s − 1.16·12-s − 0.277·13-s − 0.258·15-s + 1.03·16-s − 0.654·17-s − 0.578·18-s − 0.962·19-s + 0.899·20-s + 2.35·22-s − 0.949·23-s + 1.01·24-s + 0.200·25-s + 0.481·26-s − 0.192·27-s − 1.51·29-s + 0.448·30-s − 1.51·31-s − 0.0396·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05878270219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05878270219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 11 | \( 1 + 4.49T + 11T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 4.55T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 6.72T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 8.50T + 59T^{2} \) |
| 61 | \( 1 - 2.33T + 61T^{2} \) |
| 67 | \( 1 + 4.23T + 67T^{2} \) |
| 71 | \( 1 + 6.97T + 71T^{2} \) |
| 73 | \( 1 - 0.674T + 73T^{2} \) |
| 79 | \( 1 + 1.59T + 79T^{2} \) |
| 83 | \( 1 - 0.956T + 83T^{2} \) |
| 89 | \( 1 + 3.91T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
| 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
−7.78432709405658226647664977438, −7.24453794310597713268779728107, −6.47533139874666513946226752166, −5.90874677096595515753815478617, −5.16178395967337195597325966940, −4.28049267946770158790535386794, −3.04507228811971441867702154180, −2.08000554954511308841233222143, −1.69801463553156924294159716146, −0.14800496472388554651399243532,
0.14800496472388554651399243532, 1.69801463553156924294159716146, 2.08000554954511308841233222143, 3.04507228811971441867702154180, 4.28049267946770158790535386794, 5.16178395967337195597325966940, 5.90874677096595515753815478617, 6.47533139874666513946226752166, 7.24453794310597713268779728107, 7.78432709405658226647664977438
