L(s) = 1 | + 5·7-s + (−2.5 + 4.33i)13-s + (4 − 1.73i)19-s + (2.5 − 4.33i)25-s − 7·31-s + 11·37-s + (6.5 + 11.2i)43-s + 18·49-s + (0.5 − 0.866i)61-s + (−5.5 + 9.52i)67-s + (−8.5 − 14.7i)73-s + (−8.5 − 14.7i)79-s + (−12.5 + 21.6i)91-s + (−7 − 12.1i)97-s − 7·103-s + ⋯ |
L(s) = 1 | + 1.88·7-s + (−0.693 + 1.20i)13-s + (0.917 − 0.397i)19-s + (0.5 − 0.866i)25-s − 1.25·31-s + 1.80·37-s + (0.991 + 1.71i)43-s + 2.57·49-s + (0.0640 − 0.110i)61-s + (−0.671 + 1.16i)67-s + (−0.994 − 1.72i)73-s + (−0.956 − 1.65i)79-s + (−1.31 + 2.26i)91-s + (−0.710 − 1.23i)97-s − 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.81158 + 0.177909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81158 + 0.177909i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 5T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.5 - 11.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.5 + 14.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7 + 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.68474007339944962749448773433, −9.531899008666477722016168516955, −8.800032022198911717833949569617, −7.79072969924423516048558821470, −7.25140502158361814858796396255, −5.94290793125298037350247414372, −4.81684582862015619294128793232, −4.33426593469222670368317764669, −2.56756070212563042614493454616, −1.40988922715689211484239282205,
1.24114364036567459768051279029, 2.57980856647541561766782447009, 4.01417680841473764695521307784, 5.17408386736887837476315613948, 5.58545930878387939553856047688, 7.32098649877668305079184523340, 7.73529321176240727748756940645, 8.607142089093781397251303625625, 9.597681772198807899677412653505, 10.64629209834504164122068357952