L(s) = 1 | + 7-s + 3·11-s − 4·13-s − 6·17-s − 4·19-s − 5·25-s − 6·29-s + 2·31-s − 7·37-s + 12·41-s − 7·43-s + 6·47-s + 49-s − 3·53-s + 6·59-s + 2·61-s − 13·67-s − 9·71-s + 8·73-s + 3·77-s − 79-s − 12·89-s − 4·91-s − 4·97-s + 12·101-s − 4·103-s − 15·107-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.904·11-s − 1.10·13-s − 1.45·17-s − 0.917·19-s − 25-s − 1.11·29-s + 0.359·31-s − 1.15·37-s + 1.87·41-s − 1.06·43-s + 0.875·47-s + 1/7·49-s − 0.412·53-s + 0.781·59-s + 0.256·61-s − 1.58·67-s − 1.06·71-s + 0.936·73-s + 0.341·77-s − 0.112·79-s − 1.27·89-s − 0.419·91-s − 0.406·97-s + 1.19·101-s − 0.394·103-s − 1.45·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.779668776252674348051833997568, −7.82920805968228414292461348182, −7.07660810508611882728292760566, −6.37761234300807021530525307318, −5.46461622223372979363542208169, −4.46486608032365805907765353127, −3.93902948196541674698157974562, −2.53391337480854386395499370544, −1.72822052577214504010660043652, 0,
1.72822052577214504010660043652, 2.53391337480854386395499370544, 3.93902948196541674698157974562, 4.46486608032365805907765353127, 5.46461622223372979363542208169, 6.37761234300807021530525307318, 7.07660810508611882728292760566, 7.82920805968228414292461348182, 8.779668776252674348051833997568