L(s) = 1 | + (0.780 − 1.35i)2-s + (−0.5 + 0.866i)3-s + (−0.219 − 0.379i)4-s − 3.56·5-s + (0.780 + 1.35i)6-s + (−0.280 − 0.486i)7-s + 2.43·8-s + (−0.499 − 0.866i)9-s + (−2.78 + 4.81i)10-s + (1 − 1.73i)11-s + 0.438·12-s + (0.5 + 3.57i)13-s − 0.876·14-s + (1.78 − 3.08i)15-s + (2.34 − 4.05i)16-s + (0.780 + 1.35i)17-s + ⋯ |
L(s) = 1 | + (0.552 − 0.956i)2-s + (−0.288 + 0.499i)3-s + (−0.109 − 0.189i)4-s − 1.59·5-s + (0.318 + 0.552i)6-s + (−0.106 − 0.183i)7-s + 0.862·8-s + (−0.166 − 0.288i)9-s + (−0.879 + 1.52i)10-s + (0.301 − 0.522i)11-s + 0.126·12-s + (0.138 + 0.990i)13-s − 0.234·14-s + (0.459 − 0.796i)15-s + (0.585 − 1.01i)16-s + (0.189 + 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.772749 - 0.261532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.772749 - 0.261532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 3.57i)T \) |
good | 2 | \( 1 + (-0.780 + 1.35i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + (0.280 + 0.486i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.780 - 1.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.56 + 6.16i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + (3.78 - 6.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.780 + 1.35i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.28 + 3.95i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 0.684T + 53T^{2} \) |
| 59 | \( 1 + (-1.43 - 2.49i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.28 - 3.95i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 5.43T + 79T^{2} \) |
| 83 | \( 1 + 0.876T + 83T^{2} \) |
| 89 | \( 1 + (2.43 - 4.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.28 - 7.41i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−16.13538705210786371922051987182, −15.06129333520377956548833961225, −13.59148788467761498803386557713, −12.19312672416011351645930905074, −11.46767029879749265396053963398, −10.66784836250385217406801053823, −8.718554065492295159662662375559, −7.08999134128220321789971983341, −4.53128171992419323429389545167, −3.52701343845126251655004938817,
4.18456692599948645079505875420, 5.89823560273817359649723768672, 7.34184704771923900614729101156, 8.136387494415144605340527017090, 10.54903101242059444721771061579, 11.91949977225508773089406983661, 12.85671032012627204036292872112, 14.43100428230104194527936293766, 15.30897100800727125543660371756, 16.10483765269793869273499649411