| L(s) = 1 | + (−0.619 + 1.90i)2-s + (2.41 − 1.77i)3-s + (−3.23 − 2.35i)4-s + (0.463 − 2.62i)5-s + (1.87 + 5.70i)6-s + (4.34 + 5.18i)7-s + (6.48 − 4.68i)8-s + (2.70 − 8.58i)9-s + (4.70 + 2.50i)10-s + (18.5 − 3.26i)11-s + (−11.9 + 0.0296i)12-s + (−4.76 + 1.73i)13-s + (−12.5 + 5.05i)14-s + (−3.53 − 7.17i)15-s + (4.88 + 15.2i)16-s + (7.37 − 12.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.309 + 0.950i)2-s + (0.806 − 0.591i)3-s + (−0.807 − 0.589i)4-s + (0.0926 − 0.525i)5-s + (0.312 + 0.950i)6-s + (0.621 + 0.740i)7-s + (0.810 − 0.585i)8-s + (0.300 − 0.953i)9-s + (0.470 + 0.250i)10-s + (1.68 − 0.297i)11-s + (−0.999 + 0.00246i)12-s + (−0.366 + 0.133i)13-s + (−0.896 + 0.361i)14-s + (−0.235 − 0.478i)15-s + (0.305 + 0.952i)16-s + (0.433 − 0.751i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.50265 + 0.210741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50265 + 0.210741i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.619 - 1.90i)T \) |
| 3 | \( 1 + (-2.41 + 1.77i)T \) |
| good | 5 | \( 1 + (-0.463 + 2.62i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-4.34 - 5.18i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-18.5 + 3.26i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (4.76 - 1.73i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-7.37 + 12.7i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (28.7 - 16.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (11.3 - 13.5i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (38.2 + 13.9i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-4.43 + 5.28i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (4.65 - 8.05i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (20.9 - 7.61i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (42.0 - 7.40i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-35.0 - 41.7i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 - 25.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-15.3 - 2.70i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (48.6 - 40.8i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-11.8 - 32.5i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-23.7 - 13.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-18.2 - 31.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (43.9 - 120. i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (0.482 - 1.32i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (34.1 + 59.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (19.7 + 111. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−13.84640210046460504235702216035, −12.66728098918641612742867988842, −11.66882446540007006209208300906, −9.681630899394135580198597484303, −8.891485090606603150250708054645, −8.171893320870811739210266890295, −6.91318161149126622955372496445, −5.74898655318855485110352387087, −4.10929096054118676590217766270, −1.56799814801259263524182695463,
1.93318091871290079216383741739, 3.65531778803256158436602732827, 4.54856446920197323622427909011, 7.00658242306179632602365849334, 8.336222226783114714643418646110, 9.226623634569301027376143791271, 10.38415222249828033679380130883, 10.94775371715530151392266007707, 12.27209422153023238060409141820, 13.47858624789024027864058788285
