L(s) = 1 | − 1.58·2-s + (1.09 − 1.34i)3-s + 0.510·4-s + (−0.5 − 0.866i)5-s + (−1.73 + 2.12i)6-s + (−1.17 + 2.37i)7-s + 2.36·8-s + (−0.609 − 2.93i)9-s + (0.792 + 1.37i)10-s + (1.49 − 2.58i)11-s + (0.557 − 0.685i)12-s + (2.53 − 4.39i)13-s + (1.85 − 3.75i)14-s + (−1.71 − 0.274i)15-s − 4.76·16-s + (−3.18 − 5.51i)17-s + ⋯ |
L(s) = 1 | − 1.12·2-s + (0.631 − 0.775i)3-s + 0.255·4-s + (−0.223 − 0.387i)5-s + (−0.707 + 0.869i)6-s + (−0.442 + 0.896i)7-s + 0.834·8-s + (−0.203 − 0.979i)9-s + (0.250 + 0.433i)10-s + (0.449 − 0.778i)11-s + (0.161 − 0.197i)12-s + (0.703 − 1.21i)13-s + (0.495 − 1.00i)14-s + (−0.441 − 0.0710i)15-s − 1.19·16-s + (−0.772 − 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.321038 - 0.566273i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.321038 - 0.566273i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.17 - 2.37i)T \) |
good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 11 | \( 1 + (-1.49 + 2.58i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.53 + 4.39i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.18 + 5.51i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.69 - 6.40i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.418 - 0.725i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.82T + 31T^{2} \) |
| 37 | \( 1 + (-2.60 + 4.51i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.00 - 1.74i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.06 + 1.85i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 + (1.91 + 3.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.78T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + (-2.01 - 3.48i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + (-5.39 - 9.33i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.18 - 5.51i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.444 + 0.770i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.28323637427590660241947608108, −10.13304120319795437662015591539, −9.107041987678399868661374404159, −8.519853693665751582855407273312, −8.015562068837797469638287977877, −6.72618262966100756626912571780, −5.67307179130840698661728017676, −3.82145408867055633590340363725, −2.32594497350695249461695177720, −0.64071802180545093583663199915,
1.94655800164980277634879856093, 3.91751252415406085321863855741, 4.40780740325004054369594261969, 6.56851528376368062548901428435, 7.38903853659771802423274974140, 8.505828615618093218786572827737, 9.179081903625813966830367443639, 9.963152397856426613470047698559, 10.73410107459287960530531726966, 11.39694903442715549427032084622