L(s) = 1 | + (0.5 + 0.866i)2-s + (−1 + 1.73i)3-s + (−0.499 + 0.866i)4-s − 1.99·6-s + (2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s + (−0.999 − 1.73i)12-s − 5·13-s + (−0.499 + 2.59i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (0.499 − 0.866i)18-s + (0.5 + 0.866i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.577 + 0.999i)3-s + (−0.249 + 0.433i)4-s − 0.816·6-s + (0.755 + 0.654i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s + (−0.288 − 0.499i)12-s − 1.38·13-s + (−0.133 + 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (0.117 − 0.204i)18-s + (0.114 + 0.198i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.148962 + 1.16894i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.148962 + 1.16894i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.5 - 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 - 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−11.85115777396025630429834607139, −11.12435138854973601135057685370, −9.803782425553275554822408473598, −9.461126600388202688365852668925, −7.892467303370522582414865008916, −7.28376091097989379362930528953, −5.63867174552484529466766480318, −5.07551888602853680524476464589, −4.34418538785401174897943503141, −2.59181383908898301318882789232,
0.800446660330718816972998274858, 2.22384871379814696734719086683, 3.87286367510841135646676474849, 5.16259828207491419207351390092, 6.06264390259798833353238355358, 7.30329480218328598363271540711, 7.969805950153393282713075363887, 9.366269867331991152360444223579, 10.59679505996717267568608169242, 11.07606730156686220569036062209