| L(s) = 1 | + 8·2-s + 64·4-s + 3.41·5-s − 815.·7-s + 512·8-s + 27.3·10-s − 1.66e3·11-s + 1.17e4·13-s − 6.52e3·14-s + 4.09e3·16-s + 3.69e3·17-s − 4.68e4·19-s + 218.·20-s − 1.33e4·22-s − 2.35e4·23-s − 7.81e4·25-s + 9.36e4·26-s − 5.22e4·28-s − 2.41e5·29-s + 1.41e5·31-s + 3.27e4·32-s + 2.95e4·34-s − 2.78e3·35-s − 6.55e4·37-s − 3.74e5·38-s + 1.74e3·40-s − 3.35e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.0122·5-s − 0.898·7-s + 0.353·8-s + 0.00864·10-s − 0.377·11-s + 1.47·13-s − 0.635·14-s + 0.250·16-s + 0.182·17-s − 1.56·19-s + 0.00611·20-s − 0.267·22-s − 0.403·23-s − 0.999·25-s + 1.04·26-s − 0.449·28-s − 1.84·29-s + 0.850·31-s + 0.176·32-s + 0.129·34-s − 0.0109·35-s − 0.212·37-s − 1.10·38-s + 0.00432·40-s − 0.760·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.41T + 7.81e4T^{2} \) |
| 7 | \( 1 + 815.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.66e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.17e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.69e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.68e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.41e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.55e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.35e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.83e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.14e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.79e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.16e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.21e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.19e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.11e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.88e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.91e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.15e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.84e6T + 8.07e13T^{2} \) |
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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
−11.09378447735804532391701051092, −10.25209695729389308273831699770, −8.992082649148076703782764520841, −7.80991353526120782822471333110, −6.43794360153618000040953956643, −5.80710629165486757214179491491, −4.21515978437984462710079845274, −3.27566470018705979339971701361, −1.82151710855034396669434741580, 0,
1.82151710855034396669434741580, 3.27566470018705979339971701361, 4.21515978437984462710079845274, 5.80710629165486757214179491491, 6.43794360153618000040953956643, 7.80991353526120782822471333110, 8.992082649148076703782764520841, 10.25209695729389308273831699770, 11.09378447735804532391701051092
