| L(s) = 1 | + (−1.44 − 1.80i)2-s + (2.66 + 0.402i)3-s + (−0.745 + 3.26i)4-s + (−2.09 − 1.43i)5-s + (−3.12 − 5.40i)6-s + (−1.09 + 1.89i)7-s + (2.80 − 1.35i)8-s + (4.09 + 1.26i)9-s + (0.438 + 5.85i)10-s + (0.694 + 3.04i)11-s + (−3.30 + 8.41i)12-s + (0.257 − 3.43i)13-s + (5.00 − 0.753i)14-s + (−5.02 − 4.66i)15-s + (−0.462 − 0.222i)16-s + (−1.92 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−1.01 − 1.27i)2-s + (1.54 + 0.232i)3-s + (−0.372 + 1.63i)4-s + (−0.938 − 0.639i)5-s + (−1.27 − 2.20i)6-s + (−0.413 + 0.715i)7-s + (0.993 − 0.478i)8-s + (1.36 + 0.421i)9-s + (0.138 + 1.85i)10-s + (0.209 + 0.917i)11-s + (−0.953 + 2.42i)12-s + (0.0713 − 0.951i)13-s + (1.33 − 0.201i)14-s + (−1.29 − 1.20i)15-s + (−0.115 − 0.0557i)16-s + (−0.467 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528855 - 0.377317i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528855 - 0.377317i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.03 + 2.55i)T \) |
| good | 2 | \( 1 + (1.44 + 1.80i)T + (-0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-2.66 - 0.402i)T + (2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (2.09 + 1.43i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (1.09 - 1.89i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.694 - 3.04i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.257 + 3.43i)T + (-12.8 - 1.93i)T^{2} \) |
| 17 | \( 1 + (1.92 - 1.31i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (3.23 - 0.996i)T + (15.6 - 10.7i)T^{2} \) |
| 23 | \( 1 + (-4.82 + 4.47i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (5.34 - 0.806i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + (-0.778 + 1.98i)T + (-22.7 - 21.0i)T^{2} \) |
| 37 | \( 1 + (1.73 + 3.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.33 - 1.66i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.260 + 1.14i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.777 - 10.3i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (5.53 + 2.66i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-0.913 - 2.32i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (-9.87 + 3.04i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-8.30 - 7.70i)T + (5.30 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-0.624 + 8.33i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (4.90 - 8.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.83 + 0.427i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 2.23i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + (-0.939 - 4.11i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−15.59238184180793197579905540557, −14.87534263706378308766205917702, −12.88043839528537690857063835280, −12.35899897272329567872899411185, −10.72529677955583619361140701085, −9.394327397059009432195327302858, −8.714469437232136101159624235638, −7.81092590874933374475136101538, −3.97870418589518055973641404751, −2.50294294458104842656796662973,
3.58174178444803421268793409390, 6.76247377004281882929780867013, 7.49259483904133812086635581207, 8.589495480557880019998233813887, 9.455729013079964191248719849030, 11.11603654464193959532172599933, 13.39282381080707402361169958396, 14.35791934489161544383032104066, 15.19849791362882648860495787915, 16.06425190635400477543536383849
