| L(s) = 1 | − 3-s − 2·7-s + 9-s + 2·13-s + 8·17-s + 6·19-s + 2·21-s + 23-s − 5·25-s − 27-s + 2·29-s − 4·31-s + 6·37-s − 2·39-s + 10·41-s + 6·43-s − 3·49-s − 8·51-s + 12·53-s − 6·57-s + 4·59-s − 10·61-s − 2·63-s − 6·67-s − 69-s + 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.94·17-s + 1.37·19-s + 0.436·21-s + 0.208·23-s − 25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.914·43-s − 3/7·49-s − 1.12·51-s + 1.64·53-s − 0.794·57-s + 0.520·59-s − 1.28·61-s − 0.251·63-s − 0.733·67-s − 0.120·69-s + 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.187709168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187709168\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.77548269686906918551213797413, −9.852723889726572597553328454702, −9.327572549734856304605480844325, −7.915861682786545942527010939647, −7.22585436362211470227233056719, −5.97093962738784673352306894890, −5.50791669149893240196713363383, −4.02654482981313016099846974165, −3.00955264562177025506271471400, −1.06286256834653035747656787165,
1.06286256834653035747656787165, 3.00955264562177025506271471400, 4.02654482981313016099846974165, 5.50791669149893240196713363383, 5.97093962738784673352306894890, 7.22585436362211470227233056719, 7.915861682786545942527010939647, 9.327572549734856304605480844325, 9.852723889726572597553328454702, 10.77548269686906918551213797413
