L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s − i·6-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.500 − 0.866i)10-s + (0.965 − 1.67i)11-s + (0.258 − 0.965i)12-s + (−0.707 − 0.707i)14-s + (−0.500 + 0.866i)15-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (−0.258 − 0.965i)3-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s − i·6-s + (−0.866 − 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.500 − 0.866i)10-s + (0.965 − 1.67i)11-s + (0.258 − 0.965i)12-s + (−0.707 − 0.707i)14-s + (−0.500 + 0.866i)15-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.350951566\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350951566\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
good | 11 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 0.517iT - T^{2} \) |
| 31 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
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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
−10.69921071769570620454037681712, −9.081316243722043226804502044964, −8.305934285398508523159911657041, −7.51469827063414704455924755502, −6.56033006117860591857935078252, −6.06875994625605675910180096876, −4.98155897091201151609110596130, −3.77745853413530718105092909867, −3.04066883175567666631694681673, −1.15029544354776058309685933176,
2.38668046505405623420174342648, 3.44674353619890925242661847378, 4.17400353987416815645024420155, 4.98164597890761156367122274744, 6.30148653999987410378650283507, 6.67145462382977562213374872183, 7.84088338525734909444041803274, 9.365191150479111441356944981119, 9.839442717186369555785262473611, 10.64163122474578965857028441160