L(s) = 1 | + (0.347 − 0.347i)2-s + (0.176 − 1.72i)3-s + 1.75i·4-s + (1.16 − 1.90i)5-s + (−0.536 − 0.659i)6-s + (−0.707 − 0.707i)7-s + (1.30 + 1.30i)8-s + (−2.93 − 0.607i)9-s + (−0.256 − 1.06i)10-s + 2.67i·11-s + (3.03 + 0.310i)12-s + (2.14 − 2.14i)13-s − 0.490·14-s + (−3.07 − 2.34i)15-s − 2.61·16-s + (−3.26 + 3.26i)17-s + ⋯ |
L(s) = 1 | + (0.245 − 0.245i)2-s + (0.101 − 0.994i)3-s + 0.879i·4-s + (0.522 − 0.852i)5-s + (−0.219 − 0.269i)6-s + (−0.267 − 0.267i)7-s + (0.461 + 0.461i)8-s + (−0.979 − 0.202i)9-s + (−0.0810 − 0.337i)10-s + 0.805i·11-s + (0.874 + 0.0895i)12-s + (0.596 − 0.596i)13-s − 0.131·14-s + (−0.795 − 0.606i)15-s − 0.653·16-s + (−0.792 + 0.792i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09251 - 0.516598i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09251 - 0.516598i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.176 + 1.72i)T \) |
| 5 | \( 1 + (-1.16 + 1.90i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.347 + 0.347i)T - 2iT^{2} \) |
| 11 | \( 1 - 2.67iT - 11T^{2} \) |
| 13 | \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.26 - 3.26i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.24iT - 19T^{2} \) |
| 23 | \( 1 + (-2.54 - 2.54i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 5.28T + 31T^{2} \) |
| 37 | \( 1 + (2.14 + 2.14i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.66 + 7.66i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.43 + 4.43i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.159T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + (5.41 + 5.41i)T + 67iT^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-4.16 + 4.16i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.89iT - 79T^{2} \) |
| 83 | \( 1 + (4.03 + 4.03i)T + 83iT^{2} \) |
| 89 | \( 1 + 3.95T + 89T^{2} \) |
| 97 | \( 1 + (1.86 + 1.86i)T + 97iT^{2} \) |
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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
−13.23806933126271267357114288223, −12.73195638420862697815409659419, −12.02384622685424869074877174625, −10.61164957956366329633708048896, −8.997805312189156157665155436044, −8.134922109747898602213332364036, −7.00473770034724808110455540459, −5.56335218045376400600811067595, −3.81378935783300803668452714187, −1.95593745688978063229320285775,
2.84756299872267457114284816361, 4.62151244137868787217991599147, 5.88584433572898520220544980304, 6.78416364138023333545945051806, 8.919165530147509188709391781762, 9.616297132950294685728721646770, 10.87670398118492051583745963281, 11.22466996838569723204127017975, 13.43744941006863907659783704560, 14.02926795452803656667032623387