L(s) = 1 | + (−1 − 1.73i)5-s + (2 − 1.73i)7-s + (2 − 3.46i)11-s − 13-s + (2 + 3.46i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + (−2.5 + 4.33i)31-s + (−5 − 1.73i)35-s + (−0.5 − 0.866i)37-s − 4·41-s − 43-s + (−2 − 3.46i)47-s + (1.00 − 6.92i)49-s + (3 − 5.19i)53-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.755 − 0.654i)7-s + (0.603 − 1.04i)11-s − 0.277·13-s + (0.458 + 0.794i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + (−0.449 + 0.777i)31-s + (−0.845 − 0.292i)35-s + (−0.0821 − 0.142i)37-s − 0.624·41-s − 0.152·43-s + (−0.291 − 0.505i)47-s + (0.142 − 0.989i)49-s + (0.412 − 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444154535\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444154535\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−8.951297834981584285047186033021, −8.451468426737524779257643468464, −7.79454727188045230766938865850, −6.86659241874847083867447749607, −5.87709095707091569619648553349, −4.93163193188680856381261427790, −4.18497595772537515592320783777, −3.32394498754455384129707769689, −1.71275475893907579126943292192, −0.58954452528650424285732341348,
1.62597588473604705754291005042, 2.65606180408635787200687168914, 3.75952702627859515378801027724, 4.71741545567992137796134865458, 5.55478974079256793713211384708, 6.63363186334519995382773677609, 7.38349497597691632079330980808, 7.928349240166063164959685697382, 9.063825146922620725638963314174, 9.598965139400811704567752916848