L(s) = 1 | + 4·2-s − 16·4-s + 225·7-s − 192·8-s + 434·11-s − 613·13-s + 900·14-s − 256·16-s − 878·17-s − 731·19-s + 1.73e3·22-s + 2.85e3·23-s − 2.45e3·26-s − 3.60e3·28-s + 7.58e3·29-s + 2.17e3·31-s + 5.12e3·32-s − 3.51e3·34-s + 9.31e3·37-s − 2.92e3·38-s + 1.20e4·41-s + 1.12e3·43-s − 6.94e3·44-s + 1.14e4·46-s + 2.98e4·47-s + 3.38e4·49-s + 9.80e3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.73·7-s − 1.06·8-s + 1.08·11-s − 1.00·13-s + 1.22·14-s − 1/4·16-s − 0.736·17-s − 0.464·19-s + 0.764·22-s + 1.12·23-s − 0.711·26-s − 0.867·28-s + 1.67·29-s + 0.406·31-s + 0.883·32-s − 0.521·34-s + 1.11·37-s − 0.328·38-s + 1.11·41-s + 0.0924·43-s − 0.540·44-s + 0.794·46-s + 1.97·47-s + 2.01·49-s + 0.503·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.945634257\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.945634257\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - p^{2} T + p^{5} T^{2} \) |
| 7 | \( 1 - 225 T + p^{5} T^{2} \) |
| 11 | \( 1 - 434 T + p^{5} T^{2} \) |
| 13 | \( 1 + 613 T + p^{5} T^{2} \) |
| 17 | \( 1 + 878 T + p^{5} T^{2} \) |
| 19 | \( 1 + 731 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2850 T + p^{5} T^{2} \) |
| 29 | \( 1 - 7582 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2175 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9310 T + p^{5} T^{2} \) |
| 41 | \( 1 - 12040 T + p^{5} T^{2} \) |
| 43 | \( 1 - 1121 T + p^{5} T^{2} \) |
| 47 | \( 1 - 29878 T + p^{5} T^{2} \) |
| 53 | \( 1 - 5740 T + p^{5} T^{2} \) |
| 59 | \( 1 - 5174 T + p^{5} T^{2} \) |
| 61 | \( 1 + 38717 T + p^{5} T^{2} \) |
| 67 | \( 1 + 31707 T + p^{5} T^{2} \) |
| 71 | \( 1 + 64472 T + p^{5} T^{2} \) |
| 73 | \( 1 - 19790 T + p^{5} T^{2} \) |
| 79 | \( 1 + 105000 T + p^{5} T^{2} \) |
| 83 | \( 1 + 3318 T + p^{5} T^{2} \) |
| 89 | \( 1 - 65376 T + p^{5} T^{2} \) |
| 97 | \( 1 - 919 p T + p^{5} T^{2} \) |
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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
−11.64908290461878227556363926883, −10.60290565393575969243573971041, −9.201929699883822370174171614781, −8.538488717271853999875731614335, −7.32867951103358588653503287507, −5.99814124456197905139388728237, −4.67691313183840581374233658677, −4.38933575187365145338295646070, −2.57661229110839417530667142816, −0.991462844381091733265447790361,
0.991462844381091733265447790361, 2.57661229110839417530667142816, 4.38933575187365145338295646070, 4.67691313183840581374233658677, 5.99814124456197905139388728237, 7.32867951103358588653503287507, 8.538488717271853999875731614335, 9.201929699883822370174171614781, 10.60290565393575969243573971041, 11.64908290461878227556363926883