| L(s) = 1 | + 2-s − 4-s − 5-s − 7-s − 3·8-s − 10-s − 6·13-s − 14-s − 16-s + 4·19-s + 20-s + 25-s − 6·26-s + 28-s − 2·29-s + 5·32-s + 35-s + 6·37-s + 4·38-s + 3·40-s − 10·41-s + 4·43-s − 4·47-s + 49-s + 50-s + 6·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s − 1.06·8-s − 0.316·10-s − 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.223·20-s + 1/5·25-s − 1.17·26-s + 0.188·28-s − 0.371·29-s + 0.883·32-s + 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.474·40-s − 1.56·41-s + 0.609·43-s − 0.583·47-s + 1/7·49-s + 0.141·50-s + 0.832·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8346223549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8346223549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.88000937944593, −13.32420231170479, −12.88947689353230, −12.40300952222353, −12.05145707099593, −11.58028671581844, −11.08286351587631, −10.20604072326716, −9.876794857296465, −9.375553664830620, −9.024803619232381, −8.198429504631209, −7.821223917789506, −7.244401361935527, −6.678090052672340, −6.128574173307996, −5.395671954735817, −4.979002846899712, −4.648889982180631, −3.809623704222444, −3.485356036069146, −2.762384197811691, −2.290172217212825, −1.167753610786524, −0.2790988009244908,
0.2790988009244908, 1.167753610786524, 2.290172217212825, 2.762384197811691, 3.485356036069146, 3.809623704222444, 4.648889982180631, 4.979002846899712, 5.395671954735817, 6.128574173307996, 6.678090052672340, 7.244401361935527, 7.821223917789506, 8.198429504631209, 9.024803619232381, 9.375553664830620, 9.876794857296465, 10.20604072326716, 11.08286351587631, 11.58028671581844, 12.05145707099593, 12.40300952222353, 12.88947689353230, 13.32420231170479, 13.88000937944593
