L(s) = 1 | + (4.00 − 6.93i)3-s + (−3.98 − 6.90i)5-s + (3.19 − 5.53i)7-s + (−18.5 − 32.1i)9-s + (2.85 + 4.95i)11-s + (−12.2 − 21.1i)13-s − 63.8·15-s + (−3.04 + 5.28i)17-s + (−26.3 + 45.6i)19-s + (−25.5 − 44.3i)21-s − 67.4·23-s + (30.7 − 53.2i)25-s − 81.1·27-s + 128.·29-s + (41.5 − 167. i)31-s + ⋯ |
L(s) = 1 | + (0.770 − 1.33i)3-s + (−0.356 − 0.617i)5-s + (0.172 − 0.298i)7-s + (−0.687 − 1.19i)9-s + (0.0783 + 0.135i)11-s + (−0.260 − 0.451i)13-s − 1.09·15-s + (−0.0435 + 0.0753i)17-s + (−0.318 + 0.551i)19-s + (−0.265 − 0.460i)21-s − 0.611·23-s + (0.245 − 0.425i)25-s − 0.578·27-s + 0.822·29-s + (0.240 − 0.970i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.803673 - 1.60920i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.803673 - 1.60920i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 31 | \( 1 + (-41.5 + 167. i)T \) |
good | 3 | \( 1 + (-4.00 + 6.93i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (3.98 + 6.90i)T + (-62.5 + 108. i)T^{2} \) |
| 7 | \( 1 + (-3.19 + 5.53i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-2.85 - 4.95i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (12.2 + 21.1i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (3.04 - 5.28i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (26.3 - 45.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 67.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 128.T + 2.43e4T^{2} \) |
| 37 | \( 1 + (-119. + 206. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-4.18 - 7.24i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (0.491 - 0.851i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 513.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-302. - 524. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (229. - 397. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 - 16.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + (58.8 + 102. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (238. + 413. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-498. - 862. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-528. + 914. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (128. + 222. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 393.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 350.T + 9.12e5T^{2} \) |
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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
−12.54499649257206924918947594141, −12.05304706767360304897457263193, −10.49131401589266373177090262460, −9.023875823752931441765321271087, −8.063791441973508253431936616727, −7.40451109139463556311704838221, −6.04542695258660716805145695008, −4.23656054743975398889359044326, −2.45479044751172857923500635965, −0.901475236808122727282522924743,
2.66828833044680183568084751702, 3.86122002095395105524031930338, 5.01623428591318209391099177683, 6.77389749578702078677034396148, 8.230005605177776503649461181079, 9.122682841801348849491820614255, 10.13042931020257952751607250768, 10.99687061016984084572990179864, 12.06821738563570573573270624546, 13.66265279933536820211497285793