L(s) = 1 | + (−0.135 − 2.58i)2-s + (1.90 + 1.53i)3-s + (−4.67 + 0.491i)4-s + (2.23 − 0.0410i)5-s + (3.72 − 5.12i)6-s + (2.52 + 0.791i)7-s + (1.09 + 6.91i)8-s + (0.619 + 2.91i)9-s + (−0.409 − 5.77i)10-s + (−5.30 − 1.12i)11-s + (−9.64 − 6.26i)12-s + (−0.126 − 0.247i)13-s + (1.70 − 6.63i)14-s + (4.31 + 3.36i)15-s + (8.52 − 1.81i)16-s + (−4.22 − 1.62i)17-s + ⋯ |
L(s) = 1 | + (−0.0958 − 1.82i)2-s + (1.09 + 0.888i)3-s + (−2.33 + 0.245i)4-s + (0.999 − 0.0183i)5-s + (1.51 − 2.09i)6-s + (0.954 + 0.299i)7-s + (0.387 + 2.44i)8-s + (0.206 + 0.970i)9-s + (−0.129 − 1.82i)10-s + (−1.59 − 0.339i)11-s + (−2.78 − 1.80i)12-s + (−0.0349 − 0.0686i)13-s + (0.455 − 1.77i)14-s + (1.11 + 0.868i)15-s + (2.13 − 0.452i)16-s + (−1.02 − 0.393i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13471 - 0.993983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13471 - 0.993983i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.23 + 0.0410i)T \) |
| 7 | \( 1 + (-2.52 - 0.791i)T \) |
good | 2 | \( 1 + (0.135 + 2.58i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.90 - 1.53i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (5.30 + 1.12i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.126 + 0.247i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (4.22 + 1.62i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.00214 + 0.0204i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (5.45 - 0.285i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (0.413 + 0.569i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.64 - 3.68i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (0.0129 - 0.0199i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-5.37 - 1.74i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.35 + 4.35i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.23 - 3.21i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.34 + 2.89i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (3.75 + 4.17i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.43 - 1.28i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (5.44 - 14.1i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-1.09 + 0.796i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-6.88 + 4.47i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (1.03 - 2.31i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.74 + 1.06i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (5.48 - 6.09i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (13.8 + 2.19i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.46253456138052517869469766673, −11.16063254033729449195563196539, −10.43710894650102957753001883248, −9.743641555093470447217161478258, −8.822235795841998916332308644766, −8.150167454198919043330612851602, −5.32867216368395338399458652402, −4.37567337823825325087702383317, −2.86280890548423158072260116128, −2.13533267788473035877820132450,
2.17828828588322962171898808112, 4.62202085965418755679091781155, 5.75580982969408797052708499395, 6.91822706473938614822966653872, 7.84358829496491609723460394086, 8.324273474221903357516426020286, 9.372892291605804763568477329538, 10.56490016558438730514912643220, 12.74309291372454170075081754347, 13.53169409064087734046400567099