| L(s) = 1 | + 4·5-s − 4·7-s + 6·13-s − 6·17-s − 6·19-s − 2·23-s + 11·25-s − 2·29-s + 4·31-s − 16·35-s − 10·37-s − 2·41-s + 2·43-s + 6·47-s + 9·49-s − 12·59-s − 2·61-s + 24·65-s − 4·67-s − 6·71-s − 4·73-s − 8·79-s − 24·85-s + 6·89-s − 24·91-s − 24·95-s + 6·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.51·7-s + 1.66·13-s − 1.45·17-s − 1.37·19-s − 0.417·23-s + 11/5·25-s − 0.371·29-s + 0.718·31-s − 2.70·35-s − 1.64·37-s − 0.312·41-s + 0.304·43-s + 0.875·47-s + 9/7·49-s − 1.56·59-s − 0.256·61-s + 2.97·65-s − 0.488·67-s − 0.712·71-s − 0.468·73-s − 0.900·79-s − 2.60·85-s + 0.635·89-s − 2.51·91-s − 2.46·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.05848445303122942734898286578, −6.47078198017188932112508635672, −6.16125246919564122875288487268, −5.70549456825201821349164938171, −4.61162394909457849457108917430, −3.80324461424583610432807232534, −2.94631352416183722729126268798, −2.18961430710468708902648891412, −1.43516529245327702672189638807, 0,
1.43516529245327702672189638807, 2.18961430710468708902648891412, 2.94631352416183722729126268798, 3.80324461424583610432807232534, 4.61162394909457849457108917430, 5.70549456825201821349164938171, 6.16125246919564122875288487268, 6.47078198017188932112508635672, 7.05848445303122942734898286578
