L(s) = 1 | − 0.725i·2-s + 19.6i·3-s + 31.4·4-s + 14.2·6-s − 49i·7-s − 46.0i·8-s − 143.·9-s + 666.·11-s + 618. i·12-s − 650. i·13-s − 35.5·14-s + 973.·16-s − 1.18e3i·17-s + 103. i·18-s + 1.56e3·19-s + ⋯ |
L(s) = 1 | − 0.128i·2-s + 1.26i·3-s + 0.983·4-s + 0.161·6-s − 0.377i·7-s − 0.254i·8-s − 0.588·9-s + 1.65·11-s + 1.23i·12-s − 1.06i·13-s − 0.0484·14-s + 0.950·16-s − 0.996i·17-s + 0.0754i·18-s + 0.994·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.900512951\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.900512951\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + 49iT \) |
good | 2 | \( 1 + 0.725iT - 32T^{2} \) |
| 3 | \( 1 - 19.6iT - 243T^{2} \) |
| 11 | \( 1 - 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 650. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.07e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.29e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 5.51e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.97e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.85e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 675. iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.29e4iT - 8.58e9T^{2} \) |
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
−11.56747542842429710770708656783, −10.93215217789796996540463027422, −9.909956328465065664103438821573, −9.256989177431032117317783732331, −7.69027436718918716209788719979, −6.61659349724714654412557693828, −5.32542772183890970907090379396, −3.98023680574650254558146979086, −3.02417088929904646044511933564, −1.11996315233722222587323019420,
1.31562820103702237273734432952, 2.02188499118541663053174572586, 3.73217024238547272792880930843, 5.79060854992652268465546655804, 6.68838416238483559044529495390, 7.25308632207355758057379669385, 8.471920739833752295394099375015, 9.621901760360368506171618823393, 11.18843568944449391200932634389, 11.89973685161212664221688674724