L(s) = 1 | − 4·5-s − 4·7-s − 4·11-s − 4·13-s − 6·17-s − 19-s − 6·23-s + 11·25-s − 2·29-s − 2·31-s + 16·35-s + 4·37-s + 6·41-s − 4·43-s − 2·47-s + 9·49-s + 6·53-s + 16·55-s − 4·59-s − 10·61-s + 16·65-s − 8·67-s − 2·73-s + 16·77-s − 14·79-s − 16·83-s + 24·85-s + ⋯ |
L(s) = 1 | − 1.78·5-s − 1.51·7-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 0.229·19-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.359·31-s + 2.70·35-s + 0.657·37-s + 0.937·41-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.824·53-s + 2.15·55-s − 0.520·59-s − 1.28·61-s + 1.98·65-s − 0.977·67-s − 0.234·73-s + 1.82·77-s − 1.57·79-s − 1.75·83-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.910424209436467226777304440060, −7.39841435492544052664442274969, −6.72950423472255228560372128637, −5.81636521987769670507563277506, −4.64105699328563059546993081215, −4.09576466163470275393774902456, −3.14093156035081707278369045534, −2.43899481601303924083267001618, 0, 0,
2.43899481601303924083267001618, 3.14093156035081707278369045534, 4.09576466163470275393774902456, 4.64105699328563059546993081215, 5.81636521987769670507563277506, 6.72950423472255228560372128637, 7.39841435492544052664442274969, 7.910424209436467226777304440060