| L(s) = 1 | − 3-s − 7-s + 9-s + 3·17-s − 4·19-s + 21-s − 3·23-s − 27-s + 10·29-s + 3·31-s + 4·37-s + 4·41-s + 6·43-s − 9·47-s + 49-s − 3·51-s + 13·53-s + 4·57-s + 14·59-s − 13·61-s − 63-s − 2·67-s + 3·69-s − 2·71-s + 4·73-s + 79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.727·17-s − 0.917·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s + 1.85·29-s + 0.538·31-s + 0.657·37-s + 0.624·41-s + 0.914·43-s − 1.31·47-s + 1/7·49-s − 0.420·51-s + 1.78·53-s + 0.529·57-s + 1.82·59-s − 1.66·61-s − 0.125·63-s − 0.244·67-s + 0.361·69-s − 0.237·71-s + 0.468·73-s + 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.213764455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.213764455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.79556122156228, −12.28616137127099, −11.96092218195285, −11.57102693202265, −10.95213726234682, −10.40753263008687, −10.21421764347979, −9.708867960808715, −9.174972453725237, −8.572416059732772, −8.176651340948033, −7.667295718135452, −7.123447117850752, −6.464980468738306, −6.317022766923188, −5.702110100448807, −5.240189738478784, −4.513520950651756, −4.257017784333148, −3.612766521450917, −2.911902651802611, −2.458831767726579, −1.752211311361675, −0.9260382730906799, −0.5214658601438032,
0.5214658601438032, 0.9260382730906799, 1.752211311361675, 2.458831767726579, 2.911902651802611, 3.612766521450917, 4.257017784333148, 4.513520950651756, 5.240189738478784, 5.702110100448807, 6.317022766923188, 6.464980468738306, 7.123447117850752, 7.667295718135452, 8.176651340948033, 8.572416059732772, 9.174972453725237, 9.708867960808715, 10.21421764347979, 10.40753263008687, 10.95213726234682, 11.57102693202265, 11.96092218195285, 12.28616137127099, 12.79556122156228
