| L(s) = 1 | + (1.68 + 1.08i)2-s + (0.426 + 2.96i)3-s + (1.66 + 3.63i)4-s + (2.07 + 0.608i)5-s + (−2.49 + 5.45i)6-s + (−14.7 + 16.9i)7-s + (−1.13 + 7.91i)8-s + (−8.63 + 2.53i)9-s + (2.82 + 3.26i)10-s + (−10.4 + 6.72i)11-s + (−10.0 + 6.48i)12-s + (16.8 + 19.4i)13-s + (−43.1 + 12.6i)14-s + (−0.922 + 6.41i)15-s + (−10.4 + 12.0i)16-s + (9.81 − 21.4i)17-s + ⋯ |
| L(s) = 1 | + (0.594 + 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (0.185 + 0.0544i)5-s + (−0.169 + 0.371i)6-s + (−0.794 + 0.917i)7-s + (−0.0503 + 0.349i)8-s + (−0.319 + 0.0939i)9-s + (0.0894 + 0.103i)10-s + (−0.286 + 0.184i)11-s + (−0.242 + 0.156i)12-s + (0.359 + 0.415i)13-s + (−0.823 + 0.241i)14-s + (−0.0158 + 0.110i)15-s + (−0.163 + 0.188i)16-s + (0.139 − 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.804564 + 1.80443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804564 + 1.80443i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.68 - 1.08i)T \) |
| 3 | \( 1 + (-0.426 - 2.96i)T \) |
| 23 | \( 1 + (-43.5 - 101. i)T \) |
| good | 5 | \( 1 + (-2.07 - 0.608i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (14.7 - 16.9i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (10.4 - 6.72i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-16.8 - 19.4i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-9.81 + 21.4i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-17.4 - 38.1i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (13.9 - 30.6i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (2.29 - 15.9i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (-373. + 109. i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-347. - 101. i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (23.5 + 163. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-76.1 + 87.8i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-232. - 268. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-67.5 + 469. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-14.6 - 9.44i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (-681. - 437. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (144. + 316. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (551. + 637. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-561. + 164. i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (-84.1 - 585. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (1.09e3 + 321. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−13.15932573261615810355880272913, −12.20353270091481067810121574277, −11.20958031921269327907398589442, −9.820382723184675480646472327756, −9.036045282453950758611063067747, −7.68651184710970629025927839149, −6.24302798087879854299497999281, −5.40577328129978400550282215383, −3.92530851696404805408321495856, −2.59573769357424365571350149877,
0.821148312789546822391345146478, 2.73820207704483840011398452963, 4.06779254643155602211058806512, 5.71352705167630141765759354107, 6.74182059143912405678905487586, 7.892542519018353510947296723970, 9.407725907194698071972957306951, 10.46408820615695659065566989639, 11.37909091562878466063293985128, 12.70603093354694105024763356993
