| L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s − 2·15-s − 6·17-s + 4·19-s + 8·23-s − 25-s + 27-s + 10·29-s + 33-s + 6·37-s − 10·41-s + 4·43-s − 2·45-s + 8·47-s − 7·49-s − 6·51-s + 10·53-s − 2·55-s + 4·57-s + 12·59-s − 14·61-s + 12·67-s + 8·69-s + 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.516·15-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.85·29-s + 0.174·33-s + 0.986·37-s − 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s − 49-s − 0.840·51-s + 1.37·53-s − 0.269·55-s + 0.529·57-s + 1.56·59-s − 1.79·61-s + 1.46·67-s + 0.963·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.291684562\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.291684562\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.62675550200767, −11.99883977239098, −11.49638783601020, −11.39879265861790, −10.76765943791562, −10.20598381405693, −9.851734938786716, −9.160865723910506, −8.846394002213688, −8.494205612506613, −8.000900875903009, −7.439136940002604, −7.059751934239291, −6.658306428800758, −6.176714105384244, −5.363923338635968, −4.889955220062218, −4.440055589194590, −3.964478614340264, −3.469696810518037, −2.814525448710540, −2.558604452239864, −1.730045663762931, −0.9951440607581618, −0.5296611466256332,
0.5296611466256332, 0.9951440607581618, 1.730045663762931, 2.558604452239864, 2.814525448710540, 3.469696810518037, 3.964478614340264, 4.440055589194590, 4.889955220062218, 5.363923338635968, 6.176714105384244, 6.658306428800758, 7.059751934239291, 7.439136940002604, 8.000900875903009, 8.494205612506613, 8.846394002213688, 9.160865723910506, 9.851734938786716, 10.20598381405693, 10.76765943791562, 11.39879265861790, 11.49638783601020, 11.99883977239098, 12.62675550200767
