| L(s) = 1 | + 1.41·2-s + 2.00·4-s + 2.64i·7-s + 2.82·8-s − 5.15i·11-s − 6.04i·13-s + 3.74i·14-s + 4.00·16-s − 5.99·17-s + 10.0·19-s − 7.29i·22-s + 20.1·23-s − 8.54i·26-s + 5.29i·28-s + 26.5i·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 0.377i·7-s + 0.353·8-s − 0.468i·11-s − 0.464i·13-s + 0.267i·14-s + 0.250·16-s − 0.352·17-s + 0.529·19-s − 0.331i·22-s + 0.874·23-s − 0.328i·26-s + 0.188i·28-s + 0.915i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.455642232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.455642232\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 5.15iT - 121T^{2} \) |
| 13 | \( 1 + 6.04iT - 169T^{2} \) |
| 17 | \( 1 + 5.99T + 289T^{2} \) |
| 19 | \( 1 - 10.0T + 361T^{2} \) |
| 23 | \( 1 - 20.1T + 529T^{2} \) |
| 29 | \( 1 - 26.5iT - 841T^{2} \) |
| 31 | \( 1 + 6.40T + 961T^{2} \) |
| 37 | \( 1 + 30.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 54.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.37iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 48.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 35.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 85.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 42.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 35.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 49.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 17.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 123. iT - 9.40e3T^{2} \) |
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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
−8.579365555642615392340326639024, −7.43580034544378489199034440148, −7.01231909251524389948167155576, −5.91339088944093174371862658946, −5.47468200926484799910933141419, −4.64668752617524107664764773614, −3.64250798566413256452165909385, −2.95342116747963285705759325132, −1.98103462444741394409555064038, −0.69884719869120372494781979863,
0.964948303483395842733652594539, 2.09246326446867407013151709166, 3.03076762184683620770481321268, 3.97186459968967214062141465062, 4.66400969462114915443453714657, 5.37949462934930477223685207782, 6.37413964094622195427729210119, 6.93149437662219135284493105311, 7.66179294569049952625165897269, 8.460226172521346337960029020090
