L(s) = 1 | + 2.44·3-s + (−1.41 − 1.73i)5-s − 1.41i·7-s + 2.99·9-s − 2i·11-s + 5.65·13-s + (−3.46 − 4.24i)15-s + 4.89i·17-s − 6i·19-s − 3.46i·21-s + 7.07i·23-s + (−0.999 + 4.89i)25-s + 6.92i·29-s − 6.92·31-s − 4.89i·33-s + ⋯ |
L(s) = 1 | + 1.41·3-s + (−0.632 − 0.774i)5-s − 0.534i·7-s + 0.999·9-s − 0.603i·11-s + 1.56·13-s + (−0.894 − 1.09i)15-s + 1.18i·17-s − 1.37i·19-s − 0.755i·21-s + 1.47i·23-s + (−0.199 + 0.979i)25-s + 1.28i·29-s − 1.24·31-s − 0.852i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.78157 - 0.574892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.78157 - 0.574892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.41 + 1.73i)T \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 + 4.24iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−11.41056462141304689031755471100, −10.69294911773775605095699217584, −9.193900943157787453732950081822, −8.735474726663684743790124357829, −8.015495146484805654520699214756, −7.02422195222200638147693103363, −5.47435247954890005392237467462, −3.90293463602792371090500467811, −3.41959735746845665314996242588, −1.46150173448109023417904371919,
2.19495323186898686631424848099, 3.28437876669398359447693629076, 4.18454433343679117702411069805, 5.98948120768050121943946152919, 7.16977732142889799517297754320, 8.079073207846955296087764913964, 8.729175156178059515010123413887, 9.725865985419956438211176274897, 10.73133324948674614040552317000, 11.74714217447152739124648739026