| L(s) = 1 | + (1.20 + 0.742i)2-s + (−1.47 − 0.900i)3-s + (0.896 + 1.78i)4-s + (−0.726 + 0.165i)5-s + (−1.11 − 2.18i)6-s + (−0.411 − 0.198i)7-s + (−0.247 + 2.81i)8-s + (1.37 + 2.66i)9-s + (−0.997 − 0.339i)10-s + (2.81 − 0.316i)11-s + (0.283 − 3.45i)12-s + (0.0833 − 0.104i)13-s + (−0.347 − 0.543i)14-s + (1.22 + 0.409i)15-s + (−2.39 + 3.20i)16-s + (−2.02 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (0.851 + 0.525i)2-s + (−0.854 − 0.520i)3-s + (0.448 + 0.893i)4-s + (−0.324 + 0.0741i)5-s + (−0.453 − 0.891i)6-s + (−0.155 − 0.0748i)7-s + (−0.0876 + 0.996i)8-s + (0.459 + 0.888i)9-s + (−0.315 − 0.107i)10-s + (0.847 − 0.0955i)11-s + (0.0817 − 0.996i)12-s + (0.0231 − 0.0289i)13-s + (−0.0929 − 0.145i)14-s + (0.316 + 0.105i)15-s + (−0.597 + 0.801i)16-s + (−0.490 + 0.490i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05550 + 1.20479i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05550 + 1.20479i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.20 - 0.742i)T \) |
| 3 | \( 1 + (1.47 + 0.900i)T \) |
| 29 | \( 1 + (1.40 - 5.19i)T \) |
| good | 5 | \( 1 + (0.726 - 0.165i)T + (4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (0.411 + 0.198i)T + (4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 0.316i)T + (10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (-0.0833 + 0.104i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.02 - 2.02i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.45 - 7.00i)T + (-14.8 - 11.8i)T^{2} \) |
| 23 | \( 1 + (-8.11 - 1.85i)T + (20.7 + 9.97i)T^{2} \) |
| 31 | \( 1 + (-2.95 + 4.71i)T + (-13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (1.09 - 9.74i)T + (-36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (-5.44 - 5.44i)T + 41iT^{2} \) |
| 43 | \( 1 + (3.32 + 5.28i)T + (-18.6 + 38.7i)T^{2} \) |
| 47 | \( 1 + (-0.387 + 0.0436i)T + (45.8 - 10.4i)T^{2} \) |
| 53 | \( 1 + (-0.290 - 1.27i)T + (-47.7 + 22.9i)T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 + (-7.45 + 2.60i)T + (47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (1.56 + 1.95i)T + (-14.9 + 65.3i)T^{2} \) |
| 71 | \( 1 + (-7.43 + 9.32i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (6.72 + 10.7i)T + (-31.6 + 65.7i)T^{2} \) |
| 79 | \( 1 + (0.843 - 7.48i)T + (-77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (1.83 - 0.885i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (7.73 + 4.85i)T + (38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 9.17i)T + (-75.8 - 60.4i)T^{2} \) |
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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
−11.13253196105120054966382687194, −10.01553976223467118801508893163, −8.630801053492984952876720126504, −7.80704050553404364378397418314, −6.88157801679102516387035584882, −6.29590158885952125309840654238, −5.41217451215511115281098264110, −4.37064295694710343120738656450, −3.41275101650489778822128962499, −1.70446953831792754006760612372,
0.73766523515833895123898921629, 2.59387988601415520214381261120, 3.93653826303129438032368342120, 4.58112423236185806272082713138, 5.48049991816004289508002880461, 6.57298293410165746224632800149, 7.05318024854382974071968567301, 8.922924470248705486329688636153, 9.510299896862721200521602566759, 10.54349628435293236249836333302
