| L(s) = 1 | − 2.28·3-s + 2.23·9-s − 11-s − 0.874·13-s + 4.57·17-s + 3.16·19-s + 23-s + 1.74·27-s − 8.70·29-s − 2.62·31-s + 2.28·33-s + 0.236·37-s + 2·39-s − 9.69·41-s − 2.23·43-s − 1.41·47-s − 10.4·51-s − 1.23·53-s − 7.23·57-s + 11.1·59-s − 5.11·61-s + 7.47·67-s − 2.28·69-s + 6.70·71-s + 2.95·73-s − 11.9·79-s − 10.7·81-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 0.745·9-s − 0.301·11-s − 0.242·13-s + 1.10·17-s + 0.725·19-s + 0.208·23-s + 0.336·27-s − 1.61·29-s − 0.470·31-s + 0.398·33-s + 0.0388·37-s + 0.320·39-s − 1.51·41-s − 0.340·43-s − 0.206·47-s − 1.46·51-s − 0.169·53-s − 0.958·57-s + 1.44·59-s − 0.655·61-s + 0.912·67-s − 0.275·69-s + 0.796·71-s + 0.345·73-s − 1.34·79-s − 1.18·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.28T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 0.874T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 3.16T + 19T^{2} \) |
| 23 | \( 1 - T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 2.62T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 + 9.69T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 1.23T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 5.11T + 61T^{2} \) |
| 67 | \( 1 - 7.47T + 67T^{2} \) |
| 71 | \( 1 - 6.70T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 8.27T + 83T^{2} \) |
| 89 | \( 1 + 1.95T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.25608357342835448809849199736, −6.58805434361454850012975825082, −5.82027171619566868292604810163, −5.28575587748807214741612295194, −4.94052976760129333548520506552, −3.81018121098235062735919730276, −3.16512698255567273629083516220, −1.98776374345381751251063445401, −1.00270290093464359016966584370, 0,
1.00270290093464359016966584370, 1.98776374345381751251063445401, 3.16512698255567273629083516220, 3.81018121098235062735919730276, 4.94052976760129333548520506552, 5.28575587748807214741612295194, 5.82027171619566868292604810163, 6.58805434361454850012975825082, 7.25608357342835448809849199736
