| L(s) = 1 | + 2.45·2-s + 4.01·4-s − 1.23·5-s + 7-s + 4.92·8-s − 3.03·10-s − 2.19·11-s + 1.52·13-s + 2.45·14-s + 4.06·16-s − 7.06·17-s − 1.45·19-s − 4.96·20-s − 5.38·22-s − 2.60·23-s − 3.46·25-s + 3.73·26-s + 4.01·28-s − 6.11·29-s + 1.83·31-s + 0.101·32-s − 17.3·34-s − 1.23·35-s + 0.914·37-s − 3.56·38-s − 6.10·40-s + 1.13·41-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.00·4-s − 0.554·5-s + 0.377·7-s + 1.74·8-s − 0.960·10-s − 0.662·11-s + 0.422·13-s + 0.655·14-s + 1.01·16-s − 1.71·17-s − 0.333·19-s − 1.11·20-s − 1.14·22-s − 0.544·23-s − 0.692·25-s + 0.732·26-s + 0.757·28-s − 1.13·29-s + 0.329·31-s + 0.0179·32-s − 2.97·34-s − 0.209·35-s + 0.150·37-s − 0.577·38-s − 0.965·40-s + 0.177·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 6.11T + 29T^{2} \) |
| 31 | \( 1 - 1.83T + 31T^{2} \) |
| 37 | \( 1 - 0.914T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 2.48T + 43T^{2} \) |
| 47 | \( 1 + 0.620T + 47T^{2} \) |
| 53 | \( 1 + 6.86T + 53T^{2} \) |
| 59 | \( 1 + 8.13T + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 - 3.24T + 73T^{2} \) |
| 79 | \( 1 + 8.62T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 2.62T + 89T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
| 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
−7.36145430199954052244984908848, −6.50735530837628379782415455250, −6.01060419801298238925684065705, −5.26232972800414523653214220763, −4.49433630020797767515863894216, −4.12617297435819589623554219809, −3.34877473446048063578097169806, −2.44096605029810718843551568927, −1.80312502875150527816709895254, 0,
1.80312502875150527816709895254, 2.44096605029810718843551568927, 3.34877473446048063578097169806, 4.12617297435819589623554219809, 4.49433630020797767515863894216, 5.26232972800414523653214220763, 6.01060419801298238925684065705, 6.50735530837628379782415455250, 7.36145430199954052244984908848
