L(s) = 1 | + (0.210 − 1.71i)3-s + (0.707 − 0.707i)7-s + (−2.91 − 0.725i)9-s + 6.43i·11-s + (−4.23 − 4.23i)13-s + (1.00 + 1.00i)17-s + 6.08i·19-s + (−1.06 − 1.36i)21-s + (−0.649 + 0.649i)23-s + (−1.86 + 4.85i)27-s + 0.486·29-s − 2.38·31-s + (11.0 + 1.35i)33-s + (−4.10 + 4.10i)37-s + (−8.17 + 6.39i)39-s + ⋯ |
L(s) = 1 | + (0.121 − 0.992i)3-s + (0.267 − 0.267i)7-s + (−0.970 − 0.241i)9-s + 1.93i·11-s + (−1.17 − 1.17i)13-s + (0.243 + 0.243i)17-s + 1.39i·19-s + (−0.232 − 0.297i)21-s + (−0.135 + 0.135i)23-s + (−0.358 + 0.933i)27-s + 0.0902·29-s − 0.427·31-s + (1.92 + 0.236i)33-s + (−0.674 + 0.674i)37-s + (−1.30 + 1.02i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9477105276\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9477105276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.210 + 1.71i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 11 | \( 1 - 6.43iT - 11T^{2} \) |
| 13 | \( 1 + (4.23 + 4.23i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.00 - 1.00i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.08iT - 19T^{2} \) |
| 23 | \( 1 + (0.649 - 0.649i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.486T + 29T^{2} \) |
| 31 | \( 1 + 2.38T + 31T^{2} \) |
| 37 | \( 1 + (4.10 - 4.10i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.91iT - 41T^{2} \) |
| 43 | \( 1 + (-6.74 - 6.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.70 + 4.70i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.00 - 4.00i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.86T + 59T^{2} \) |
| 61 | \( 1 + 7.72T + 61T^{2} \) |
| 67 | \( 1 + (8.66 - 8.66i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.938iT - 71T^{2} \) |
| 73 | \( 1 + (-0.399 - 0.399i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.65iT - 79T^{2} \) |
| 83 | \( 1 + (-8.35 + 8.35i)T - 83iT^{2} \) |
| 89 | \( 1 - 8.32T + 89T^{2} \) |
| 97 | \( 1 + (-1.26 + 1.26i)T - 97iT^{2} \) |
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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
−9.298757003632200067839187263810, −8.093006840231711165256288921102, −7.68991564527635945851219939946, −7.15561400144462268221494321272, −6.20866862359636628162820922824, −5.30095172568591204895094609383, −4.50592292235921691130378913740, −3.26397762704286928724901818477, −2.22535527207621750259892128548, −1.38695463217357277846556988898,
0.32012583266378801599020079583, 2.26371551356318032987116510085, 3.12703620522925790736677688412, 4.04763771796536789626918463469, 4.96510292906180545245645487832, 5.55804840113677023010397289622, 6.51232690043924982630511333878, 7.47915682806546844219769306156, 8.457742206440698140506090905561, 9.073104288261139460976728559615