| L(s) = 1 | + (0.766 − 0.642i)3-s + (0.826 + 0.300i)5-s + (1.09 − 1.89i)7-s + (0.173 − 0.984i)9-s + (0.0812 + 0.140i)11-s + (0.581 + 0.487i)13-s + (0.826 − 0.300i)15-s + (0.539 + 3.05i)17-s + (2.77 − 3.35i)19-s + (−0.379 − 2.15i)21-s + (−1.21 + 0.441i)23-s + (−3.23 − 2.71i)25-s + (−0.500 − 0.866i)27-s + (−1.13 + 6.41i)29-s + (−0.479 + 0.829i)31-s + ⋯ |
| L(s) = 1 | + (0.442 − 0.371i)3-s + (0.369 + 0.134i)5-s + (0.412 − 0.715i)7-s + (0.0578 − 0.328i)9-s + (0.0244 + 0.0424i)11-s + (0.161 + 0.135i)13-s + (0.213 − 0.0776i)15-s + (0.130 + 0.741i)17-s + (0.637 − 0.770i)19-s + (−0.0827 − 0.469i)21-s + (−0.252 + 0.0920i)23-s + (−0.647 − 0.543i)25-s + (−0.0962 − 0.166i)27-s + (−0.210 + 1.19i)29-s + (−0.0860 + 0.149i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46628 - 0.374742i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46628 - 0.374742i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-2.77 + 3.35i)T \) |
| good | 5 | \( 1 + (-0.826 - 0.300i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.09 + 1.89i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0812 - 0.140i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.581 - 0.487i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.539 - 3.05i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (1.21 - 0.441i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.13 - 6.41i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (0.479 - 0.829i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + (8.11 - 6.81i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.166 - 0.0605i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.602 - 3.41i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (7.83 - 2.84i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (0.482 + 2.73i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.79 - 2.47i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.184 + 1.04i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (4.77 + 1.73i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.72 + 1.44i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (4.01 - 3.36i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.55 + 14.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.9 - 10.0i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (1.63 + 9.27i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.22192585230608836323643538082, −11.13528075356730028292517747305, −10.22395695387569029891360712677, −9.190254788975328938386995932241, −8.108987370415782040791965205877, −7.20373027391651151396261184950, −6.13380945525004501970131775081, −4.67459349767027361157900609009, −3.28360601466425857467801560194, −1.60618311660037267623317490885,
2.04929780112509732001629821488, 3.54099697526627965675639475883, 5.02841446485067326199502239102, 5.93770865525428713565115711654, 7.49708124632241816953790461648, 8.456735322700669445700837608852, 9.405808399335044550146203135498, 10.16538255585450685874559023501, 11.43638897968202561136695797399, 12.17599049037329134164513689728
