| L(s) = 1 | + (1.30 − 1.51i)2-s + (−2.52 + 1.62i)3-s + (−0.569 − 3.95i)4-s + (−1.00 − 2.19i)5-s + (−0.853 + 5.93i)6-s + (19.8 − 5.83i)7-s + (−6.73 − 4.32i)8-s + (3.73 − 8.18i)9-s + (−4.63 − 1.36i)10-s + (−20.6 − 23.8i)11-s + (7.85 + 9.06i)12-s + (−12.3 − 3.62i)13-s + (17.2 − 37.6i)14-s + (6.09 + 3.91i)15-s + (−15.3 + 4.50i)16-s + (15.3 − 106. i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (−0.0896 − 0.196i)5-s + (−0.0580 + 0.404i)6-s + (1.07 − 0.315i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.146 − 0.0430i)10-s + (−0.565 − 0.652i)11-s + (0.189 + 0.218i)12-s + (−0.263 − 0.0772i)13-s + (0.328 − 0.719i)14-s + (0.104 + 0.0673i)15-s + (−0.239 + 0.0704i)16-s + (0.219 − 1.52i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.983177 - 1.32183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.983177 - 1.32183i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.30 + 1.51i)T \) |
| 3 | \( 1 + (2.52 - 1.62i)T \) |
| 23 | \( 1 + (45.2 - 100. i)T \) |
| good | 5 | \( 1 + (1.00 + 2.19i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (-19.8 + 5.83i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (20.6 + 23.8i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (12.3 + 3.62i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-15.3 + 106. i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (21.9 + 152. i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (0.911 - 6.34i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-225. - 144. i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (140. - 307. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (-24.1 - 52.8i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-198. + 127. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 31.7T + 1.03e5T^{2} \) |
| 53 | \( 1 + (466. - 137. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (128. + 37.5i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-597. - 384. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-406. + 469. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-494. + 571. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-117. - 819. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-310. - 91.0i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-193. + 423. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (825. - 530. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (-643. - 1.40e3i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−12.17531795381494265197793152577, −11.37888458504993281143178854028, −10.71331205849171476691602512453, −9.520204651284686003797952740973, −8.240020337482600111654983824224, −6.84241351582548320224115988516, −5.18729993469179969509740324010, −4.65320828473974468030705248245, −2.84622887704185121838872235410, −0.78205613456047798753299790028,
1.97790421526970211964567908672, 4.11872494169277268124433127231, 5.32782791525415977513080342746, 6.33124580288320708634879909645, 7.71418300640160168668382774047, 8.319002056662552133203185030235, 10.12902167000215941152198362698, 11.11286474299391792178775971487, 12.31732691100109457103312141789, 12.76010940273228683139978068228
