L(s) = 1 | − 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (−4.99 − 0.134i)5-s − 2.44·6-s + 11.0·7-s + 2.82i·8-s − 2.99·9-s + (−0.190 + 7.06i)10-s − 9.44i·11-s + 3.46i·12-s + 22.3i·13-s − 15.6i·14-s + (−0.233 + 8.65i)15-s + 4.00·16-s − 29.3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (−0.999 − 0.0269i)5-s − 0.408·6-s + 1.58·7-s + 0.353i·8-s − 0.333·9-s + (−0.0190 + 0.706i)10-s − 0.858i·11-s + 0.288i·12-s + 1.72i·13-s − 1.11i·14-s + (−0.0155 + 0.577i)15-s + 0.250·16-s − 1.72·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.035962209\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035962209\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + (4.99 + 0.134i)T \) |
| 23 | \( 1 + (20.6 - 10.0i)T \) |
good | 7 | \( 1 - 11.0T + 49T^{2} \) |
| 11 | \( 1 + 9.44iT - 121T^{2} \) |
| 13 | \( 1 - 22.3iT - 169T^{2} \) |
| 17 | \( 1 + 29.3T + 289T^{2} \) |
| 19 | \( 1 - 31.8iT - 361T^{2} \) |
| 29 | \( 1 - 21.8T + 841T^{2} \) |
| 31 | \( 1 + 33.9T + 961T^{2} \) |
| 37 | \( 1 + 7.59T + 1.36e3T^{2} \) |
| 41 | \( 1 - 55.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 77.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 70.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 21.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 31.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 64.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 8.93iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.63T + 6.88e3T^{2} \) |
| 89 | \( 1 - 153. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 71.0T + 9.40e3T^{2} \) |
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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
−10.90935103041555903173562090363, −9.293274555536873090329916989176, −8.464683198822388973455794290084, −7.986703695451025253838782626000, −6.97252783554078983205932585487, −5.75711049142294260441113283820, −4.43013795598046933046057428706, −3.95264283772433981647751054429, −2.25965975453649043331063290373, −1.32709928360181286719707455074,
0.39766401090453667352496867252, 2.49171376489582598304413268602, 4.11659377505025540703514582945, 4.68069608912717377085442180130, 5.43135620967813572848324393542, 6.91959865048112232698682624427, 7.64148712469090120385245721871, 8.431444379104631483403587293051, 8.973641341893672575999622472451, 10.38119173706612629315642482419