| L(s) = 1 | − 16·2-s + 256·4-s − 625·5-s − 1.32e3·7-s − 4.09e3·8-s + 1.00e4·10-s + 1.55e4·11-s + 6.19e4·13-s + 2.11e4·14-s + 6.55e4·16-s − 8.61e4·17-s − 1.09e6·19-s − 1.60e5·20-s − 2.48e5·22-s + 1.16e6·23-s + 3.90e5·25-s − 9.91e5·26-s − 3.38e5·28-s + 2.10e6·29-s + 6.75e6·31-s − 1.04e6·32-s + 1.37e6·34-s + 8.26e5·35-s − 5.83e6·37-s + 1.75e7·38-s + 2.56e6·40-s + 7.84e6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.208·7-s − 0.353·8-s + 0.316·10-s + 0.319·11-s + 0.601·13-s + 0.147·14-s + 0.250·16-s − 0.250·17-s − 1.93·19-s − 0.223·20-s − 0.226·22-s + 0.867·23-s + 0.200·25-s − 0.425·26-s − 0.104·28-s + 0.553·29-s + 1.31·31-s − 0.176·32-s + 0.176·34-s + 0.0931·35-s − 0.511·37-s + 1.36·38-s + 0.158·40-s + 0.433·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 16T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 625T \) |
| good | 7 | \( 1 + 1.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.55e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.19e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 8.61e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.09e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.16e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.10e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 6.75e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.83e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.84e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.04e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.82e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.51e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.24e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.34e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.73e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.75e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.25e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.76e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.34e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.48e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.14e8T + 7.60e17T^{2} \) |
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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
−9.857923912404343806079949714807, −8.697281006675285896700313962519, −8.237799662597435513635198670166, −6.87647645988225772485277380384, −6.27684615465212726535091887123, −4.72088344733412756272383673414, −3.58277272798495067968565536348, −2.35126353764829641208866623948, −1.08891934602421180735616614476, 0,
1.08891934602421180735616614476, 2.35126353764829641208866623948, 3.58277272798495067968565536348, 4.72088344733412756272383673414, 6.27684615465212726535091887123, 6.87647645988225772485277380384, 8.237799662597435513635198670166, 8.697281006675285896700313962519, 9.857923912404343806079949714807
