| L(s) = 1 | + (−1.68 − 1.08i)2-s + (0.426 + 2.96i)3-s + (1.66 + 3.63i)4-s + (15.3 + 4.49i)5-s + (2.49 − 5.45i)6-s + (7.46 − 8.62i)7-s + (1.13 − 7.91i)8-s + (−8.63 + 2.53i)9-s + (−20.8 − 24.1i)10-s + (−17.4 + 11.2i)11-s + (−10.0 + 6.48i)12-s + (21.0 + 24.3i)13-s + (−21.8 + 6.42i)14-s + (−6.81 + 47.3i)15-s + (−10.4 + 12.0i)16-s + (17.6 − 38.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.0821 + 0.571i)3-s + (0.207 + 0.454i)4-s + (1.36 + 0.402i)5-s + (0.169 − 0.371i)6-s + (0.403 − 0.465i)7-s + (0.0503 − 0.349i)8-s + (−0.319 + 0.0939i)9-s + (−0.660 − 0.762i)10-s + (−0.479 + 0.307i)11-s + (−0.242 + 0.156i)12-s + (0.449 + 0.518i)13-s + (−0.417 + 0.122i)14-s + (−0.117 + 0.815i)15-s + (−0.163 + 0.188i)16-s + (0.251 − 0.551i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.56110 + 0.467866i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56110 + 0.467866i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-106. + 27.9i)T \) |
| good | 5 | \( 1 + (-15.3 - 4.49i)T + (105. + 67.5i)T^{2} \) |
| 7 | \( 1 + (-7.46 + 8.62i)T + (-48.8 - 339. i)T^{2} \) |
| 11 | \( 1 + (17.4 - 11.2i)T + (552. - 1.21e3i)T^{2} \) |
| 13 | \( 1 + (-21.0 - 24.3i)T + (-312. + 2.17e3i)T^{2} \) |
| 17 | \( 1 + (-17.6 + 38.6i)T + (-3.21e3 - 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-45.8 - 100. i)T + (-4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (-42.1 + 92.3i)T + (-1.59e4 - 1.84e4i)T^{2} \) |
| 31 | \( 1 + (36.7 - 255. i)T + (-2.85e4 - 8.39e3i)T^{2} \) |
| 37 | \( 1 + (92.4 - 27.1i)T + (4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-307. - 90.4i)T + (5.79e4 + 3.72e4i)T^{2} \) |
| 43 | \( 1 + (-19.9 - 138. i)T + (-7.62e4 + 2.23e4i)T^{2} \) |
| 47 | \( 1 + 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-181. + 209. i)T + (-2.11e4 - 1.47e5i)T^{2} \) |
| 59 | \( 1 + (254. + 294. i)T + (-2.92e4 + 2.03e5i)T^{2} \) |
| 61 | \( 1 + (-96.4 + 670. i)T + (-2.17e5 - 6.39e4i)T^{2} \) |
| 67 | \( 1 + (-86.8 - 55.8i)T + (1.24e5 + 2.73e5i)T^{2} \) |
| 71 | \( 1 + (985. + 633. i)T + (1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (178. + 390. i)T + (-2.54e5 + 2.93e5i)T^{2} \) |
| 79 | \( 1 + (418. + 483. i)T + (-7.01e4 + 4.88e5i)T^{2} \) |
| 83 | \( 1 + (-28.2 + 8.30i)T + (4.81e5 - 3.09e5i)T^{2} \) |
| 89 | \( 1 + (142. + 994. i)T + (-6.76e5 + 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-627. - 184. i)T + (7.67e5 + 4.93e5i)T^{2} \) |
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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
−12.82326891290336392945043182038, −11.44717866302426668653376505199, −10.47085143027875182403509621985, −9.887331552720955157322617338253, −8.945584481828751815166810227272, −7.60776411112036847064969701908, −6.26335175237610361153086529477, −4.88426861717747035690219762953, −3.12297002034721364364370300233, −1.61500268105955931472176717920,
1.13788556821130448555392760226, 2.54177669241496702245932173042, 5.28612165635628105339575425685, 5.95759833537859959218847680655, 7.30290866375561372652112407089, 8.540218299796198351801394940382, 9.245109119611863724151092098785, 10.41194123690039908067672846184, 11.47784955663486453459555141974, 12.95991085004152883785806111076
