| L(s) = 1 | + (1.20 + 0.740i)2-s + (−2.50 − 1.75i)3-s + (0.901 + 1.78i)4-s + (−3.25 − 0.284i)5-s + (−1.71 − 3.96i)6-s + (3.39 + 1.96i)7-s + (−0.236 + 2.81i)8-s + (2.17 + 5.96i)9-s + (−3.70 − 2.75i)10-s + (−1.03 + 3.85i)11-s + (0.871 − 6.05i)12-s + (0.894 + 1.27i)13-s + (2.63 + 4.88i)14-s + (7.64 + 6.41i)15-s + (−2.37 + 3.22i)16-s + (−5.68 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (0.851 + 0.523i)2-s + (−1.44 − 1.01i)3-s + (0.450 + 0.892i)4-s + (−1.45 − 0.127i)5-s + (−0.701 − 1.61i)6-s + (1.28 + 0.741i)7-s + (−0.0834 + 0.996i)8-s + (0.723 + 1.98i)9-s + (−1.17 − 0.870i)10-s + (−0.311 + 1.16i)11-s + (0.251 − 1.74i)12-s + (0.248 + 0.354i)13-s + (0.705 + 1.30i)14-s + (1.97 + 1.65i)15-s + (−0.593 + 0.805i)16-s + (−1.37 − 0.502i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.566850 + 0.724507i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.566850 + 0.724507i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 - 0.740i)T \) |
| 19 | \( 1 + (-3.08 - 3.08i)T \) |
| good | 3 | \( 1 + (2.50 + 1.75i)T + (1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (3.25 + 0.284i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-3.39 - 1.96i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.03 - 3.85i)T + (-9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.894 - 1.27i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (5.68 + 2.07i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (1.39 - 1.66i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 2.32i)T + (-18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (0.924 - 1.60i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.81 + 3.81i)T + 37iT^{2} \) |
| 41 | \( 1 + (-3.68 + 0.650i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.93 - 0.431i)T + (42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.524 + 0.191i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-0.253 - 2.90i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-4.04 - 8.68i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (5.15 - 0.451i)T + (60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-2.87 + 6.17i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (1.59 + 1.90i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (12.2 - 2.15i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (0.339 + 1.92i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.49 - 5.57i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-7.90 - 1.39i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.96 - 1.44i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.04654975638670699966920453096, −11.51667072169083688845486969402, −10.90551386018148268858186709417, −8.624379966884534271191257932245, −7.52512693671401798565238855592, −7.31191866098944969137095241187, −5.98236809438365914926584986636, −4.96795652048983017891967315692, −4.33535466543724208024897489251, −2.01575928488030533376167713640,
0.62317354883141536999662845582, 3.52061678741107629686031397912, 4.38145499177705958279648296409, 4.97943997897041297285238459355, 6.14131764100536187576138116373, 7.33837111045534329547650301817, 8.661431942987682344835866677440, 10.32296749076498284544186850733, 11.05557161891076629359130737812, 11.22526019263937027754624217093
