| L(s) = 1 | − 7-s − 3·11-s + 4·13-s + 2·19-s − 3·23-s − 9·29-s + 8·31-s − 5·37-s + 6·41-s − 11·43-s + 6·47-s + 49-s + 6·53-s − 10·61-s − 5·67-s − 15·71-s + 10·73-s + 3·77-s − 7·79-s + 12·83-s + 12·89-s − 4·91-s − 8·97-s − 18·101-s + 10·103-s − 12·107-s − 7·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.904·11-s + 1.10·13-s + 0.458·19-s − 0.625·23-s − 1.67·29-s + 1.43·31-s − 0.821·37-s + 0.937·41-s − 1.67·43-s + 0.875·47-s + 1/7·49-s + 0.824·53-s − 1.28·61-s − 0.610·67-s − 1.78·71-s + 1.17·73-s + 0.341·77-s − 0.787·79-s + 1.31·83-s + 1.27·89-s − 0.419·91-s − 0.812·97-s − 1.79·101-s + 0.985·103-s − 1.16·107-s − 0.670·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.72672102224842722721739463016, −7.02837159632024119183112881375, −6.14767296896479487785368970427, −5.67793050505098464675587394806, −4.84358962205196989014192587584, −3.91696852791896464354634357949, −3.25499960838365532570557202720, −2.35968089857306704369133770493, −1.30109246464009715852061432732, 0,
1.30109246464009715852061432732, 2.35968089857306704369133770493, 3.25499960838365532570557202720, 3.91696852791896464354634357949, 4.84358962205196989014192587584, 5.67793050505098464675587394806, 6.14767296896479487785368970427, 7.02837159632024119183112881375, 7.72672102224842722721739463016
