L(s) = 1 | + 6.63i·2-s + 19.8i·3-s − 12·4-s + (45 − 33.1i)5-s − 132·6-s + 132. i·8-s − 153·9-s + (220. + 298. i)10-s + 252·11-s − 238. i·12-s + 119. i·13-s + (660 + 895. i)15-s − 1.26e3·16-s + 689. i·17-s − 1.01e3i·18-s + 220·19-s + ⋯ |
L(s) = 1 | + 1.17i·2-s + 1.27i·3-s − 0.375·4-s + (0.804 − 0.593i)5-s − 1.49·6-s + 0.732i·8-s − 0.629·9-s + (0.695 + 0.943i)10-s + 0.627·11-s − 0.478i·12-s + 0.195i·13-s + (0.757 + 1.02i)15-s − 1.23·16-s + 0.578i·17-s − 0.738i·18-s + 0.139·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.244685329\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.244685329\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-45 + 33.1i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 6.63iT - 32T^{2} \) |
| 3 | \( 1 - 19.8iT - 243T^{2} \) |
| 11 | \( 1 - 252T + 1.61e5T^{2} \) |
| 13 | \( 1 - 119. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 689. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 220T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.43e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.39e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 198T + 1.15e8T^{2} \) |
| 43 | \( 1 + 417. iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 5.82e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.46e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.36e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 5.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.09e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.18e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 9.99e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5iT - 8.58e9T^{2} \) |
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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
−11.61492400488104118989051106695, −10.66938773720813264628867435279, −9.501933812801325290737587307117, −9.084618099219475019334436097832, −7.907116925196730367800649140759, −6.62287456979023731019525533561, −5.62008010739812713787098889479, −4.90495009888309641192672677415, −3.69624693012619572428665713867, −1.76357753476231496978551879186,
0.60659669344056432807751757857, 1.77659665108182358239221945194, 2.45071780976177305158568266000, 3.76229697439406926303936169665, 5.68701279270296160301571207986, 6.79333607068090278460773611092, 7.35837584168313849192961712444, 8.990206537984244331858018170330, 9.872722327872960041377385898405, 10.89878479553880040152770734111