| L(s) = 1 | + (−4.5 + 2.59i)5-s + (−3.17 + 5.49i)7-s + (8.17 + 4.71i)11-s + (9.84 + 17.0i)13-s + 1.90i·17-s − 4.69·19-s + (−8.17 + 4.71i)23-s + (1 − 1.73i)25-s + (−2.84 − 1.64i)29-s + (20.5 + 35.5i)31-s − 32.9i·35-s − 17.3·37-s + (53.5 − 30.9i)41-s + (0.477 − 0.826i)43-s + (12.2 + 7.05i)47-s + ⋯ |
| L(s) = 1 | + (−0.900 + 0.519i)5-s + (−0.453 + 0.785i)7-s + (0.743 + 0.429i)11-s + (0.757 + 1.31i)13-s + 0.112i·17-s − 0.247·19-s + (−0.355 + 0.205i)23-s + (0.0400 − 0.0692i)25-s + (−0.0982 − 0.0567i)29-s + (0.662 + 1.14i)31-s − 0.942i·35-s − 0.467·37-s + (1.30 − 0.754i)41-s + (0.0110 − 0.0192i)43-s + (0.259 + 0.150i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.002064423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.002064423\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (3.17 - 5.49i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.17 - 4.71i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-9.84 - 17.0i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 1.90iT - 289T^{2} \) |
| 19 | \( 1 + 4.69T + 361T^{2} \) |
| 23 | \( 1 + (8.17 - 4.71i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (2.84 + 1.64i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-20.5 - 35.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 17.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-53.5 + 30.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-0.477 + 0.826i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-12.2 - 7.05i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 9.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-79.2 + 45.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (37.5 - 65.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-15.4 - 26.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 85.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (76.1 + 43.9i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 41.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 - 83.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.259194048593712687149113751549, −8.899520608950360891105715732607, −7.947246411682917124997718494866, −6.98118695231615993984429084785, −6.51154284064419864179825496595, −5.58327428287216713753709257361, −4.28168045833499819299540584045, −3.77041957431160221570469650991, −2.67677052810280284563197267216, −1.49580066109554924616755529604,
0.31374660764053104110659780511, 1.09271089670763644727626470940, 2.85423161508242778806935909218, 3.87407344999795321826835196252, 4.26462106656825015524961676917, 5.57259696143950180652037805993, 6.32682482926181371060231435979, 7.26654241204159889011395763101, 8.085339873370891692572286424889, 8.531221518532827374158424288434
